. ■4.:i !^H^ . "^ß^"^ »rtr '"^P"^"'' ^rv..^4 .jgr*t^^«^- ;^:, ..'v, y-. .4^ Alt^ » v> * •*t> -,> ■|.-*£.k-,;'.< :>v yF J^r^^ /.^ BIOMETEIKA A JOURNAL FOK THE STATISTICAL STUDY OF BIOLÜGICAL PROBLEMS EDITED IN CON8ULTAT10N WITH FRANCIS ÜALTON BY W. F. R. WELDON KARL PEARSON AND C. B. DAVENPORT LIBRARY ^nW YORK BOTANlCAL ÜAltDEN VOLUME II November 1902 to November 1903 CAMBRIDGE AT THE UNIVERSITV PliESS LONDON: C. J. CLAY AND SONS, AVE MARIA LANE AND H. K. LEWIS, GOWER STREET NEW YORK : THE MACMILLAN COMPANY LEIPSIC : BROCKHACS BOMBAY AND CALCUTTA : MACMILLAN AND CO., LIMITED [All Ri())ttA rcst:rvcd\ CONTENTS. Memoirs. PAGE I. Oll the Systematic Fitting of Curves to Observations and Measnre- ments. Part II. By Kaiu, Pearson, F.R.S 1 II. Quantitative Study of the Effect of Environment upon the Forms of Xassa obsoleta and Xuxsa trivittuta froni Cold Spring Harbor, Long Island. By Abigail Camp Di.mon 24 III. ün the Ambiguity of Mendel's Categories, By \V. F. R. Weldox, F.RS U IV. Cooperative Investigatioiis on Plants : I. On Inhiiitaiice in the Shirley Poppy .......... 56 V. On the Number and Arrangement of the Bony Plates of the Young John Dory. By L. W. Bykxe 11.') VI. Notes on the Theory of Association of Attributes in Statistics. By G. Udny Yule 121 VII, A Further Study of Statistics rehxting to Vaccination and Siiiall- pox. By \V. R. Macdonell 135 VIII. Cooperative Investigations on Phmts : II. Variation and Correlation in Lesser Celandine l'roni divers localities . . . .145 IX. Second Report on the Result of crossing Japanese Waltzing Mice with European Albino Races. By A. D. Dakbishire . 165 X. New Tables of the Probability Integral. By \V. F. Siieppakd . 174 XI. Variation in .£^H/)0(7H7-HÄ Pnrfea (M'ii Heller. By E. H. J. Sciu'steu 191, Xll. The Lixw of Ancestral Heredity. By Karl Pearson . .211 Appendix II. On Inheritance (Grandparent and Offspring) in Thoroughbred Racehorses. By Norman Blanchard . . 229 Appendix III. On Inheritance (Great-Grandparents and Great- Grcat-Grand])arents and Offspring) in Thoroughbred Race- horses. ]iy Alice Lee 234 Contents iii PAGE XIII. Actinosphaerium Eichurni A Biometrical Study in tho Mass Ri'liilicilis of NucliMis aiul Cytoplasin. \',y GeoFFREY ÖMITII . i-il XIV. A Proliiriinary Atteinpt to ascertain tho llelationship between size of Coli and sizo of Body in Daplniin VKtcjnd Strauss. By E. Wakhun 2.";5 XV. Graduation and Analysis of a Sickuoss Table. B}' W. Palin Elderton 2üO XVI. On tlie Probable Errors of Frequency Constants. (Editokial.) . 273 XVII. Third Report on tlie Hybrids between Waltzing Mice and Albino Races. ün the result of Crossing Japanese Waltzing Mice with " Extracted " Recessive Albinos. By A. D. Darbishire . . 282 XVIII. Mr Bateson's Revisions of Mendels Thoory of Heredity. By W. F. R. VVeldon 2SÖ XIX. Mendels Laws and sonie Records in Rabbit Breeding. By F. A. Woods 299 XX. Ueber Asymmetrie bei Gelasiniiis puyilutor Latr. Von Georg DUNCKER 307 XXI. Variation and Correlation in Arcella vulgaris. By Raymond Pearl and Frances J. Dunbar 321 XXll. On the Laws of Inheritauce in Man. I. Inheritauce of Physical Characters. By Karl Pearson 857 XXIII. Variation in Ophiocoma Nigra 0. F. Müller. By C. D. McIntosh 463 XXIV. Tables of Powers of Natural Numbers and of Sums of Powers of Natural Numbers from 1 — 100. By W. Palin Elderton . 474 XXV. Assortative Mating in Man. A Cooperative Study . . . 481 Miscellanea. (i) Note on the Results of Crossing Japanese Waltzing Mice with European Albino Races. By A. D. Darbkshire .... 101 (ii) Interpolation by Finite Differences. (Two Indopendent Variables.) By W. Palin Elderton lOö (iii) Variation in tho Muscatel {Adoxa Moschatellina, L.). By Henry Whitehead 108 (iv) Seasonal Change in the Characters of Aster prenanthoides Muhl. Note on a paper by G. H. Shull 113 iv Contents PAQE (v) Note on the Influence of Changc of Sex on the Intensity of Heredity. By FuANK E. Lutz 237 Craniological Notes: (vi) Professor von Törük's Attack on the Arithinelic Jlean. By K. PearsüN 339 (vii) Homogeneity and Hetcrogeneity in Collections of Crania. By K. Pearsox 345 (viii) Preliniinary Note on Interraoial Chamcters and their Corre- lation in Man. By S. Jacob, A. Lee and K. Pearson 347 (ix) Inlieritance of Finger Prints 356 (x) Inheritance in Phaseoliis viifmris. By W. F. K. W. and K. P. . 499 (xi) Addendniii to " Oraduation and Analysis of a Sickness Table." By W. Pamx Ei.derton 503 Craniological Kotes : (xii) Homogeneity and Hetcrogeneity in Crania. By Charles S. Myers 504 Reniarks on Dr Myers' Note. By K. Peau.sox . . 506 (xiii) On Cranial Types. By Professor Aurel von Török . 508 Remarks on Professor von Török's Note. By K. Pearson 509 Plates. Frontispiece. Francis (Jaltox, froni a photogiajili, with sketch. Presented to Biometnka by E. B. Piate I. Young John Dory, showing position of anterior dorsal and anal plates to face p. 120 Subject and name indices will be isstied every feiv years embracing several volumes. Vol. II. Part I. November 1902 BIOMETEIKA A JOURNAL FOR THE STATISTICAL STUDY OF BIOLOGICAL PROBLEMS EDITED IN CONSULTATION WITH FRANCIS GALTON BY W. F. R. WELDON KARL PEARSON AND C. B. DAVENPORT CAMBRIDGE AT THE UNIVERSITY PRESS LONDON : C. J. CLAT AND SONS, AVE MARIA LANE AND H. K. LEWIS, GOWEK STEEET KEW YORK: TDE MAOMILLÄN OOMPANT LEITSIC : BB0CKHAÜ8 BOMHAT AMD OALCUITA : MACUUXAN AND 00:, LIMITED Price Ten Shillings net. \Xttwi December 19, 1902] LINNAEA Institute of Natural History, lnvaliden=Strasse 105 BERLIN, N. 4. Greatest stock in Preparations, Collections and Models of the whole sphere of ZOOLOGY, COMPARATIVE ANATOMY, PALAEONTOLOGY, MINERALOGY, GEOLOGY AND BOTANY. Attention is particularly called to our preparations prepared in Alcohol and mounted In cyllnder-glasses, of whicb the followlng Is a selection : SITUS PREPARATIONS showing the structure of the internal organs visible in thcir natural position. Moi decumantiB Muk 3a00 Loxift curriroetra , 13-W KmTH earopaea 14.00 Uosrts viridU 12.00 Peliu berus 15 00 Bads fortls 1200 Salun&ndra macaloea •• 9 00 TiDCS Tulfcaria . . 13.00 HeUx pomatia , . • ?■"* Anodont« anatina ^ 1 00 Bombyz mori >• 6-00 AxUcus fluviatilis d-OO PREPARATIONS OF NERVES showing the course of the chief ncrvea. Hu8 decnmanus Columba lirla dorn. Bana (nrtU Tinea vulgaris . . Torpedo naroe Astacun BuvIatiUs Dytlscue margioalU Mark 2S.00 ,. 25.00 „ 1500 „ 2000 ., 30.00 >. »■- ,. 6.50 DOUBLE INJECTIONS. The arterics are injpctcd in red, and the veins in blue ; all orgnns arc visible. Bdnrus vulgaris Mark 60.00 Colombs llvia dorn „ 50 Oo LacerU oceUats 5O.0O PMudoput apoa '<"I0 Zameniii vlridlOaTU« 5U0O Ew» lociu« 50 00 SIMPLE INJECTIONS. Mus decumanos Mark 22.00 Columba Uvia dorn 25.00 20.00 20.00 Kaox lucius .. 25.00 Helti pomalla .. lOOO Astacus fluviatUis 9.00 METAMORPMOSES IN ALCOHOL. Laicrta viviiwra Mark 12.00 Anguis fraglus 15.00 Ptilias b«rus 16.00 Hyla arborea .. 12.00 Pelobatefl ruscus „ 30.00 Alytes obstetricans 30.00 üufo clnereus ... 12.00 BombinMor igncua „ 15.00 !(). maruloea . ..10 00 Aniblj-Htonia mexicamuii K.oa Klioileti!« ainaruH 20.00 tia.it*'n>steus aculcatuif ,. 16.00 l'aluitina vivipara ,. 12.00 Venpa crabro „ 13.00 Microgaster glomeratus, witb paradt« . .. 10.00 Cimbex variabdis do. ,. 15.00 HydrophduK picfus „ 20.00 Meloloiithft vulgaris „ 25.00 (-'titonfu auraU . . ,. 25.00 Krgate« faber .... ., 1500 Khynchonhorus scbacb OaclDUs Wt . . . 1500 aoo (iastroplillu» iwcorum „ 1500 I>o. tNjul „ 1500 HyiHMU-rTjia 'Unna ., 15.00 Hilf. . IN ,. 1500 Crvl.i 5.00 (Jrj; - .u. ., 1000 Ten.,.- i..,,.M- .. „ 35.00 Nei« i'inerua . . 600 t'arciuus inaenas 14.00 Lluulus |>olrphemU8 . . 15.00 In addition to the above preparations attention is called to our Collections of Mimicry and Dimorphism. Please ask for our Catalogues. Volume II NOVEMBER, \'.H)-2 Ni>. 1 BIOMETEIKA. ON THE SYSTEMATIC FITTING OF CURVES TO OBSERVATIONS AND MEASUEEMENTS. l.iERAF.V PART II.* .;. :a By KARL PEARSON, F.R.S., University College, London. CONTENTS. (9) On the Fitting of Normal Frequency Distribiitions wlien a part only of thc Frequency is given ........ Illustration V. American Trotting Horses .... (10) On the Moments of Trapezoidal Areas (11) On the Fitting of Parabolic Cm-ves of any order to given Data (a) Method of Least Squares (12) (6) Method of Moments (13) Illustration VI. Thiele's Patience Statistics .... (14) Illustration VII. Italian Marriage Statistics .... (15) General Conclusions ......... Paus 1 2 7 9 9 12 16 20 21 (9) Illustration V. In the previous illustration we have fitted the best curve not to the ordinates, but to the logarithms of the ordinates. This method was forced upon us by the complexity of Makeham's formula. It will clearly give goüd resiilts in many cases where it inight be difficult to calculatc the momcnts of the ordinates of the curve, but in whioh the raoments of the lotrarithnis of the ordinates follow quitc easily. For example in curves of the type y = e-'^'-^' we shall often introduce considerable simplicity into nur work without loss of practical efficiency by fitting to the data a curve Y=f(,r), where F=logy. In particular * Part I. was publishcd in Vol. i. p. 265. Biometrika ii 1 2 On the Si/stematic Fitthoj of Ciirres i{ f(x) he an algebraical cxprcssion in iiitcf^er powci-s of x, \vc rcducc the fitting of curves of thi.s type tu the theory of panibola-Htting with which wc shall be occupied later. A very intercsting case of such work arises in dealing witli frequency distributions, which wc suppose to bc normal or approxiniately normal, but of which only a portion of the distribution can bo known or observed. For example, the marks of cjxndidates in a competitive examiuation, wherein candidutes below a ccrtain grade have been rejectcd by a preliminary examinatiou, or are cast out without placing. Or again, the statures of the soldiera in a rcgiment with a minimum admissible height. Clearly in such cases as these we have to fit a curve givcn a certain nuniber only of values of y and x. The method of least Squares or that of moments would enable us theoretically to determine y,. « and b and so to find the constants of the best fitting normal distribution. But with tho curve in its above form tlic e(|uations, especially in the case of least scpiares, become unmauagcable. If, however, we write y = e'', wc find the problem reduces to fitting Y=a'a^ + b'x + c, wherc Y is known for a ccrtain ränge of values of x. As far ;is I know the first attempt to determine the constants of a normal curve when only a portion of the distribution is known w:is made by Mr Francis Galton in his memoir on the speed distribution of Americjvn Trotting Horses*. The American record contains only horses which oun trot a mile in less than a given number of seconds. Henco assuming the distribution to be normal we obtain oniy a portion of the frequency distribution, i.e. the number of horses that can trot a mile in each number of seconds less than this maximum. Taking a normal curve Mr Galton has determined the position of the mode, i.e. the valuc of h, only by inspection of the plottcd figures. It seemed worth wiiile to compare his rcsults with what we sliould get by fitting curves Y= a'x' + h'x + c' to the logarithms of his frequency data, using the method of fitting parabolas of the second onier discussed on ]>. 14 below-f. It seems well to briefly indicate the process used. The curve for the year 1893 was determined by mc, tho.se for 1802, 18f)4 and 18!).") I owe entircly to the encrgy of Mr Leslic Bramlcy-Moorc. * li. S. Proc. Vol. 02, p. 310. + Tho curve bcing parnbolio the methods of moments and of Icast Squares are now sensibty identical in result, altbough uot alike in their stages. K. Pkarson 3 Taking Mv Galton's polygoii lor Lliu data of liS!)2 niiiutccn ordiiiatcH were obtaiiied f'or si)eeds at equal intervals of a sccond, 29 — 28, 28 — 27 ... 11 — 10 fVoin the observations on 1324 trottei'S. On an arbitrary scale these nincteen ordinates are given in Column (1). In Coliiinii (2) are their logarithms to three figure.s. Cohinin (.')) gives tliu (irst niDuieiil ?/ti aboiit the middle of the ränge 21 = 19. Column (4) the second nioiiunt m.. about the sanio poinl. Froiu ?/*„, iHi, in. and /, Xo, Xi and X. wcre foutid and hent-e c,,, e, and e.j (see p. 14). (1) ('-') (3) (i) y 1' X S{.ry) S (x"-Y) 92-8 1-9G8 -9 -17-712 1.59-408 100-4 2-002 -8 -16-016 128-128 95-0 1-978 -7 - 13-846 96-922 71-2 1-852 -6 -11-112 66-672 67-6 1-830 -5 - 9-150 45-750 61-3 1-787 -4 - 7-148 28-592 61-4 1-788 -3 - 5-.3ß4 16-092 44-8 1-651 -2 - 3-.302 6-604 44-5 1-648 1-661 1-584 -1 + 1 - 1-G48 1-648 45-8 -85-298 38-4 1-584 1-584 27-8 1-444 + '2. 2-888 5-776 19-8 1-297 + 3 3-891 11 673 10-7 1-029 -t-4 4-116 16-404 15-8 1-199 + 5 5-995 29-975 7-9 •898 + 6 5-388 32-328 5-0 -699 + 7 4-893 34-251 2-1 -322 + 8 2-576 20-608 5-6 •748 + 9 6-732 60-588 OTo = 27-385 + 38-063 -85-298 ««2 = 763-063 mj= -47-235 Thus 2/„= '-1^; = 1-441,316, X, = -^ = - •181,.5G4, X„= '"■^, = •308,74.54. " mj- Whence e„ = r092,205, e, = - -.544,692, e., = - -276,0143. These give us for the required parabola : 7=1-441,316 j 1-092,20.J- -544,692 m --276,614 (^^ This may be thrown into the furin : F= 1-441,316 1 1-360,583 - -276,614 f* "J '''^' where x = — 9-3575. 1—2 4 On thc Sy.stematic Fitting of Ciirves Now l is 9'5 ; hence the centre of thc normal curve is at "1425 beyond the Start of the ränge wliii-li is 29 secs., i.e. 2S-8Ö75 secs. is the modal speed of the group of trotters. If the normal curve bc y = z„e Whence we find and ultimately 1 = log 2/ = log z, - ^^/ löge. log5„= 1-441,:31() X 1-300,583, 2<7= = /-'löge l-441,31(Jx-27(i,G14' ^0 = 9-14176, 0- = 7011,073. Thus thc requircd normal curve is 2/ = i)14l70e-5-'''('''"-ö'-'')'. The "probable error" corresponding to the above valuc of o- is 4"7289, which enables us to compare our resuits directly with Mr Galton's numbers. Likc Mr Galton wo have omitted from considcratioii the group of horses with spceds betwccn 29 and 30 secs., for it incltidos a large nuniber of doubtful trotters, whose speed is aliowod by grace to fall within the 30-second limit. The following are the actual determinations for four years of the 'centre' and variability of American trotters. They are coini)arcd with Mr Galton's determina- tions by inspection of the 'centre' and his calculatiou from this of the probable error by quai'tiles. Tear Constant Moments Galton 1892 Centre S.D. P.E. 28-8575 7-0111 4-7289 29 4-25 189S Centre S.D. P.E. 28-0120 8-8345 5-9588 26 5-0 1894 Centre S.D. P.E. 25-6256 6-2946 4-2456 27 4-5 1895 Centre S.D. P.E. 28-0119 8-5469 5-7641 27 5-0 K. Pbarson — 1 1 1 1 1 1 1 »> 1... 1 /•»^ > 1 „ -\ ""^V ; \ """< i ^N ^ ^ 1 ■•\ ;-■'> 1 \--. 'v '. N \ > ^^ t N \> o^ ; •^ ■-^ -^^ -^ « ■■"ö • — if 1 s e 4 1 3 1 9 e' • e « 3 a ) 10 Seconds Fio. 5. Distribution of Speed of American Trotters, 1892. 1324 Observations. 1 ' A ? / ^\ ä r ■^ J 1, ^ Ifi h- ■V»— — V r-^ \ V \ ■'•-. V **^ 'S^ ^^ A •s \/ \ v. -^ .^ /\ V. .;-^ *^* 1 s g 3 » 1 s • B • fl S ■ t 0. « 1 l \ B 1 t l i ■■ la 1 ' Seconds Fig. 6. Distribution of Speed of American Trotters, 1893. 982 Observations. Moments - - . - Inspection and Quartile On the Sifsfeiiiafic Fittiiig qf Curves to \ \ A 1" i .> > V •:.■._ \ r\ • \ \) v N ■•O,- V -> "-.'; v>\ ••^■^:: N ^^ ._ -- ... ■■ u n u u 11 I« U U II 10 • • T Fio. 7. Distribution of Speed of American Trotters, 1894. 1204 Observations. ! t ! 1 «0 r. 1 f ^ A 1/ \^ ^ ^^•i ^'•. VJ >. J. ^ M ^^ N *» \ S-: .■^ ■^ ^^ ^ ^ : ^ -^ -**' — • ■ • "" * • * i > Secondi Fio. 8. Distribution of Speed of American Trotters, 1895. 1124 Observations. Munients - ■ ■ ■ luspcction and Quartilu K. I*KARSON 7 A ghiiico ;it thc aeeoinpanying Figs. 5, ü, 7 aiid 8 will show tluit thc lit has been nmch improved by adopting thc systcmafcic method of momcnts. At thc sarae time using a Brunsviga for all niultiplications and divisions and a Compto- nictor for additions the labour is not very severe. Of course it is not contcnded tbat tbis accuracy is nccessary in the present case ; Mr Galton's approximations are probably close enough for tbe ends he had in view. We have only uscd his data to illustratc a niothod, whicb may be of scrvico for special cascs, wherc thc best avaihible deterininations of the constants are needed. (10) In calculating the moments in the previoiis illustration we have simply concentrated along thc mid-ordinates. This was close enough for the pur|)ose of illustration. When the ordinates of a curve which rcrpiires fitting are truo ordinates, say for exaraple, measurements obtained by ob.servation, theii- irregu- hirity is often such that it appears idle to use complex quadrature formulffi. Such formulas give very good results, if the ordinates are those of a mathe- matical forniida, or if we have a fairly smooth System of points. But a very frequent case is a case like that of the acconipanying figure, in which a quad- / y. V. Vm-i c ,,.--'Vo Vi rature formula for the moments seems idle and yet we are scarcely justified in concentrating along the ordinates to find the moments, if the base unit be not indefinitely small. Here it seems reasonable to take the area and moments of the System of trapezia as fairly representing the area and moments of the curve to be fitted. It would be idle to use a formula like Simpson's, for exaniple ; because the changes in curvaturo in the curve, which would be ailowed for by Simpson's method of striking a parabola through three successive points, have clearly no existence in the general sweep of the observations and are due only to irregularities of Observation. Accordingly we want an exprcssion for the moments of a systcni of trapezia. Let the ordinates he y^, yi, y^... y^ and the corresponding abscissas .%, «j, x^ ■•• a;,«. 8 On the Systeinatic Fitting of Ctirves Wc shall limit ovir discussion to equal base elemeuts b, whioh caii ahvays be takeu as uuity. Let v,:=S{x"y). Then the following expression for thc »th moment A/„' of thc trapezoidal area about the axis of y is easily deduced : ,., ,. «.(n-1) , n(»-l)(n-2)(«-3) ., «(»-l)(»-2)(n-3)(»-4)(»-.5) + — 2ÖT6Ö '^"-'^ "^••• ■^"1(2'' |3 " ^ [4 " ■ Hei-e the " corrective terms" in ^„, and y„ are nothiug niore thau thc subtraction of Ihc ni\\ monients of the triangles PQR and STV from the »th moiiicnt of the whule figure oii base RS, which is represented by the remainder of thc expression. They can bc thrown iiito the simple forms (a;„ + 6)"+' - {n + 2) a;^"+' h - x-J'^^ ^"' (n + !)(« + 2) _ (- a-, + 6)»^' - (n + 2) (- ^o)»+' fe - (- x,)"*' ^ ' ^° (?^+l)(n + 2) Now let PT= 21 and let us takc moments about the middle of PT. Let Jtf„ be the nth moment abont this point, and x being measured from it, let m Vn = S{x''y). We havc x,n = —x„ = l, and accordingly, if / bc measured in b as imit, « (h -1) «(«-!)(»- 2 )()t - :$) M„ = I'„ + -^ 1 „ — I'n-a H US?; "«-4 w(w-l)(»-2)(w-3)(w-4)(«-5) + 20160 ">.-« + (n + l)(« + 2) t^^^^^ ^^ ^'''• The corrective term is thus very simple : il may be written •in(i/m+(-l)"»/o). K. Pkarson 9 whcre La iiiay l)c calculatril micu l'or all f'or /; = to 5 aiiil t'or valucs of Z = .S to 20 or tor all odd munbers of orclinates fnnii 7 to 41. 'I'hose values are giveii in thc accoiiipaiiyiiig Taliio, wliicli will eiiable tliu ruadcr to readily find the correctivo terui, bcariiig in inind tliat, if the nionient is even, />„ mu.st be muitiplied by ym + yo> and that if it is odd it niust be muitiplied by i/m—>/o- Table of Currective Terms fov Moments of Trapezia. Ln = (n +!)(« + 2) l Lo ii ^■: I', lu L, 3 •5 1-667 5-583 18-800 63-033 216-524 4 •5 2-107 9-417 41-050 179-500 787-357 5 •5 2 -667 14-250 76-300 409-307 2200-857 6 •5 3107 20-083 127-550 811-233 5167-024 7 Ti 3-667 26-917 197-800 1455-100 10715-857 S ■5 4-167 34-750 290-050 2422-967 20257-357 9 •5 4-667 43-583 407-300 380S-833 35041-524 10 ■5 5-167 53-417 552-550 5718-700 59218-357 11 •5 5-667 64-250 728-800 8270-567 93897-857 12 ■5 6-167 76-083 939-050 11594-433 143210-024 13 ■5 6-667 88-917 1186-300 15832-300 211364-857 U •5 7-167 102-750 1473-550 21138-167 303312-357 15 •5 7-667 117-583 1803-800 27678-033 424802-524 16 •5 8-167 133-417 2180-050 35629-900 582445-357 17 T) 8-667 150-250 2605-300 45183-767 783770-857 IS ■5 9-167 168-083 3082-550 56541-633 1037289-0-24 19 •5 9-667 186-917 3614-800 69917-500 1352549-857 20 •5 10-167 206-750 4205-050 85537-367 1740203-357 (11) 0» the fitting of Pavabolic Gurves to given Data. Let US consider this probleni first froni thc standpoint of the niethod of least sijuares. Let the parabola be of the (m — l)th order and ropresented by y = Co + Ci X + c, X- + . . . + c,i_, *-"-'. Let tis write for brevity Vr = S (x'l/), X/ = .S' (*■'■), where S denotes a sumniation extending to the values of x and >/ for every Observation. Then proceeding to make S (y — Ca— Ci X — C.X- — ... — Cn-i *'" ')" Biometrika ii 10 Oll Ihr Systemalic Fittiiig of Ciirves a iiiiiiiiiiuin for values of c», Ci, C5...c„_i we casily find the type ciiuations f,' = c„X,' + c,V + CqV + . . . + c„_,\'„ i// = CoX./ + CiX/ + CjX/ + ... + c„_,X'„+, l» n_l — CjX „_, + Cj X „ + CjX ,n-i + ... +Cn_iX5„_i. These are n equations to find the n constants. Now in any curvc-fitting with 11 constants to bc found wc must detcrminc the area and nionients v„', Vi, v^ ... "',1-1 ! but we find that the inethod of Icast Squares iuvolves also the discovery of the 2h qnantitics Xo', X,', Xj' ... X'j„_,. If as is very usually the case the observations are takcn at cqual intervals and there are m of them, wc can by choosing this interval as our unit and a proper origin, write X/ = :£ (./■■•) = r + 2-- + s-- + . . . + Mi--, or X/ is the siini vi' the rth powers of the first m natural numbers. Thus the values of X/ for .suecessive values of »rt can be tabled once and for all. This is done for 7/i.= l to 20 and ?• = 1 tu 7 in the accompanyiiig 'l'able. Table of the Sinns of the first seven Powers of the Natural Numbers*. Up to S{x) .S (X') S(x3) S{.^) SM S (X«) S(.r') 1 1 1 1 1 1 1 1 3 5 9 17 33 65 129 s 6 14 36 98 276 794 2316 4 10 30 100 354 1300 4890 18700 5 15 55 225 979 4425 20515 96825 6 21 91 441 2275 12201 67171 376761 7 28 140 784 4676 29008 184820 1200304 8 36 204 1296 8772 61776 446964 3297456 9 45 285 2025 15.333 120825 978405 8080425 10 55 385 3025 25333 220825 1978405 18080425 11 66 506 4356 39974 381876 37499C6 37.567596 12 78 650 6084 60710 630708 67359,')0 73399404 13 91 819 8281 89271 1002001 11. '.62759 1.36147921 U 105 1015 11025 127687 1539825 19092295 241.561425 15 120 1240 14400 178312 2299200 30482920 412420800 16 136 1496 18496 243848 3347776 47260136 680856256 17 153 1785 23409 327369 4767633 71397705 1091194929 18 171 2109 29241 432345 6C57201 105409929 1703414961 19 190 2470 36100 562666 013.3.300 152455810 2597286700 20 210 2870 44100 722666 12333.300 216455810 3877286700 * Mr W. Palin Klderton lias calcnlated an ext«nded table of this kind, giving the sums of the 7th powur» of all the natural numbers up to 100. I hopo it miiy bc possible eventually to rcproducc and distribute this much eulargcd table. I'karson 11 ll will l)c soeii at onec lluit \~' is very largo l'or any uuiiibcr of obsurvalioiis greatiT tliaii 10. Even if \ve go as f'ar as X/, \ve shall, liowevcr, be ouly fitting a paiabola <>(' the Üiird order, and tho four type oquations to be solved will b(; found as a rule evcn in tliis case to be ratlier unmauageable. For parabolas of the fourth, fifth and sixth Orders, the labour bccomes very severe. It is clcar liowcvur that if \ve fvaluatcd tlie (h/tiTniiaaiit X„ , X, , X._. , ... X „_j X, , Xo , Xj , .. . X „ \' < ^;i > '^11 ... X „-I-, X „ ^ /i > 1^ n + 1 1 X ..,), and its iniiiors for valucs of /* =1, 2, ... •"), (i, and of m froni 1 iip to 20, wo should have a set of constants which wonld enablc u.s readily to find Co, Ci, ... as soon as the Vo, Vi,...v'n^i were given. The arithmetical work to be once done would be considerable, but it might be worth doing, .supposing the method of inoments were not available with a simpler Solution. It niay bc notod that a considerable simplification of the least Square type equation.s can be introduced if there be an odd numher of observations. Lct us take the origin at the niiddle Observation, then cloarly X, = X3 = Xj = . . . = 0, or all the odd x sums vanish. Lot ns use undashed letters to denote moments about the centre of the ränge, then we find our system of type equations breaks up into two i'o = CuX,i + c.X,^ + CjXj + and v, = CiX. + C3X4 + CsX^ + Vi = CoXo + C2X4 + CjXs + i'3 = C1X4 + c^Xß + CsXg + Vi = C„\i + cAß + CjXg + V', = Ci\ + CjXa + CsXio + Our Table will now enable us to find X^ for an)- number of observations up to 41; and for parabolas of the third order only, we have simply to solve two sets of linear equations, each of which contains only two unknowns. Thus the work becomes extremely simple. This is in faet how the cubical paraboJa was fittcd by the method of least Squares to the observations in our illustration in § 3 (Vol. i. p. 280). On the other hand even with this choice of origin we require X„, X,o and X,, to fit parabolas of the fourth, fifth and sixth orders ; and the sum of the lOth or 12th powers of the natura! numbers* and simultaneous equations with three or four variables and coefificients of very diverse niagnitudes are rather tronblcsome matters to deal with. Ol course X,. may be caIculateJ from the Bernoulli mimber formula on ji. 286, Diumetiikn, Vol. i. 2—2 12 On Ihe Systimatic Fitting of Cnrves Of courso if the \ determinant and its minore were once worked out, say for the first six parabohw and tor tlie ciistoinary ränge of values of m, we should have no niore labour, but meanwhile it seenis to nie that the method of least Squares miist for practical piirposes be laid on one side even for jiarabolic curvcs oxcept in the jiiinple cases of those of the tiret, second anil tiiird orders. But if the method of least Squares be of smail practical use even in this one of the simplest cascs of curvo-fitting, it may be (piestioned whether it is not better to adopt the uniform process of monients thruugliont. (12) Lct US now apply the method of monients to the parabolic curve y = e„ + e, a; + ßj ar* + . . . + e„_,a:"~' for which the expansion by Maclaurin's Theorem is exact. Let 21 be the ränge for which this curve is to be fitted to the observations, and let us take the origin at the mid-point of the ränge. Let m„ be the area and ?«,, nu ... 7»„_i the first n moments of the Observation poIygon about the axis of y, i.e. the perpendicular to the ränge at its mid-point. Let US write nto = 2lxy^,so that y„ is the mean value of the ordinate. Then the curve to be fitted mav bc writtcn in the form y = y«(eo + e,'''^+e.('jj +... + e„_, [j] j iMiilt iply by f -T J and iutegrate from x = l to .t= - l : l-(-l)" + ... + — ^ — ^ 2r+S 2 2r + n If we multiply by («//)"■+' and iutegrate, l+(-l)" --/^-''^' = 2i/»^{27T-3 + 24-5^--^ 2r + n + 1 It is obvious that the even e's will be givon in terms of the even moments and the odd e's in terms of the odd monients by tucj independent series of cquations. Let US write \, = mg:(7)ij'), thus X« = 1. Then X,= 60 + ^6^ + ^,6, + ... \ = ieo + iei + ie,+ ... K = },eo + \ei + ijet + ... ^^5 = 761 + ie3 + -^et+ ... K. I'kahson 13 Hencc it, is vU':iv tliat aiiy e caii 1«' expressed iit oucu in terms of Üiu iiioments and of DUO or otlier of the dettTiuinant.s 1, l/.S, l/ö. 1/7, ... 1/:J, 1/5, 1/7, 1/1), ... l/ö, 1/7, 1/9, I/Il,... Iß, l/ö, 1/7, 1/0, ... 1/5, 1/7, 1/9, 1/11,... 1/7, 1/9, 1/11, 1/18,... und l-lieir respective niinors. I( is thus (juite easy to oxpross the general result of working with e«, «i ... e,,-,. But as a matter of practica! ai)plication, it wonld involve far too much troublesomc arithmetie to calciilato momeiits beyond the tiftli or .sixth. We have accordiiigly only to caiculate once and for all the numerical coeEficients of the X's in the; values of the e's for the first few cases and these will serve for all future applications. Ca.se (i). To fit a straight liiie to a series of observations. Let the linc be y = yjeu + e, 'j Then Xg = e,,, X., = ^e,. Thus the cfjuation of the line is y = y« (\, + 3x, I GeoTnetncal Gonstriiction. Let the broken line ^jB be the observations and A'B' the best straight line. Then A'B'EF and ADBEF must have the same first moments and the same area. Let CK be the vertical through the centroid of the observations, i.e. obtained by taking their mean FK. Now the trapezium may be considered as made up of two triangles A'EF and A'B'E, the centroids of which lie in the vertical lines G^Hj^ and G.,H„ trisecting FE. Hence the area A'B'EF acting in CK must be equivalent to the areas A'EF and A'B'E, or l X A'F and l x B'E, acting in G,H, and G„H.,. Now A'F+ B'E is known, for it equals S^/o, the double of the mean Ordinate of the observations. Take Ol = 2yo and from any poiiit 0, draw 00 to meet G^H^ and CK in t and u. Draw tiv parallel to Ol to meet G,H., in v, and then draw 02 parallel to tv to meet Ol in 2. We shall then have 02 = A'F and 21 = B'E, the required lengths, which fully determine the line A'B'. The construction given is the familiär graphical one for finding the components in the lines G^H^ and G.H.. parallel to CK of a force 2^„ acting in CK. The prin- ciple of moments would also give a Solution. Thus take moments about //j : 2/ A'Fx^ = -2y„xH,K, On the Systematlc Fitting qf Curves B'E=^''xH,K, wliich dülonniiie the iiitercepts siiice JIJ{ aiiil JIJ{ are knuwn. 14 whence similarly ..• B /^' -r^'^^ -, >B ^ ■■:i:^ Ac <.■■■ A' it' 1 \ JGj ..-■■' "">-.. *=■ y °"r « ... f ".i .■■■' .- < :Hs .--'^''-, 1 .'' ,.- ■ = «P'-SJ- . ..-+'' -•'' i ..■■•■' /' _,.-[''' 6' {-'-'' ! Gase (ii). To fit a parabola of the second order to a seines of observations. Let the parabola be y = yo\eo + e,-i + e.Jjj>. Oiir cqiiations are now \, = le,. Hence e« = ^ (SX« — öXg), e, = 3a,, , K. PKARf^ON 15 Thus e„, e, , e™ aro at oncc foiimi, wlion X„, X, and X, arc known. Probably tlie best way to constnict this parabola is to draw it graphically tliroiigh thc three points *'i = -l, 2/1 = 2/0 (e» - ei + e.,), a^3 = 0, !/■> = y„e„, a^a = ^, 2/3 = 2/0 (e» + e, + e,). Gase (iii). Tofit ajmrahola oftlie ihird order to a series of ohservations. Lot thc curve be 2/ = 2/0 1^0 + e, * + e, (^* j + «3 (^1 Thc equatioiis to find thc es will now bo \. = «0 + 3^2, Xi=^e, + ^63, ^2 = 3C„ + ^e,, X3 = 5^1 + -feg. Hence Co = f (3\„ — 5X.,), e, = ^- {ö\ — TX,), e, = J^ (3X, - X„), e, = ^■^- (- 3X, + SX,)- Gase (iv). Tofit a parabola of the/ourth order to a series of ohservations. Let thc curvc bc Thcn X„= e„ + ^e, + ^64. \ = iei + ic3, ^ = 5ßo + ie.> + fe4, X3 = iei + }e3, Hence we find e„ = ^ä (1 5x„ - 70X, + 03X4), e, = -L'i (5X, - 7X3), «2 = W (- ö^^o + 42X, - 45X4), e, = ^{- 3X1 + 5X3), ei = ^ (3X„ - 30X, + 35X4). Gase (v). To fit a parabola ofthefifth order to a series of ohservations. Let the curvc be y = y„ je„ + e. ^ + e._ i^J + e. (^ + e. g)^ + e. (f )] . Tben X„ = e,+ ^e^ + ^e^, X, = Je, + ig, + fe^, ^2 = 5e„ + ie, + |e4, X3 = ie, + 1^3 + ^e.,, 5^4 = ie„ + }e, + 1^4, X5 = f e, + igj + yV^s- 16 On the Si/stemafic Fitfing qf Curves Hence we have eo = H (15X„-70X,+ 63XJ, e, = }^{- (35\,- 12GX, + DOX»), e, = J^(-5X,+ 42X,-45X4), e, = äj^^ {- 2l\+ OOX^-TTX,), e, = y{- (:}\, - 30^. + 35X,), es = -7j''- (15X,- TOX, + G3X,). Gase (vi). To fit a parabola of the sixth order to a series qf observations. Let the curve be y = y» jeo + e, f + e, [ff + e, (?)' + e. (f )' + e, (f)' + e, (f )] . Then Xo= ^„ + ^^3+ ie^ + ^e,, X, = ^e, + ^63 + fßs, ^! = Jeo + ^ej+ |e4 + ^e„ X, = ^e, + |e, + ^gj, X-4 = ieo + jej+ ^e^+^V^e, X5 = |e, + ^«3 + 1^65, ^ = f e„ + ^63 + jVe« + tV««- Hence we find eo = ih (3öX„-315X,+ 693X,- 429X,), e, = Vr (35X, -126X, + 99X,). «! = 5i5(-35X„ + 567X, - USÖX, + lOOlX,), e, = ^«-(- 21X. + 9OX3 - TTX,), «4 = ^s'W^ (7X„-135X.,+ 3S5X4- 273X,), 65 = W (l-'>^.- 70X3 + 63X5), e6 = 4i"^(-5X«+10.5X,- 31ÖX,+ 231X6). (13) Illustration VI. In order to thoroughly tost the manner in which succes- sive parabolas fit niore and niore closcly to a series of observations, I have taken as a first illustration a very unproniir^ing series of observations given by Thiele in his Foi'elaesninger over Almindeltg lagttagelsealaere (K^ifihcnhiivii, 1.S89), p. 12. I say u)ipromising because the observations are not such sis one wouid in practice attempt to fit with a parabolio cuive; thcy form a frequcncy distribntion for whieh my skew frequency curve of limited ränge gives a very good fit as we have seen above in § 6. But to fit thcse unproniising observations evcn approxiniately is of great interest; the process shows us much inore clearly than would otherwise be the case the struggles of the successive parabolas to get their points of inöexion to the approxiniately corrcct pnsitions, and, to sjjcak anthroponiorphicall}', their rather futile attenipts to bend theniselvcs into the shape of the Observation curve. But a still morc important principle is illustratcd whcn we compare ihese p.arabülas with the geueralised frequency curve, naniely that the nunibcr of constants at our dispo.sal is no ineasure of the goodness of the fit. The skew frequency curve with three coustants fits much better than the parabola of the sixth Order with seven constants. Thus in fitting an empirical curve to observations it is all-important to make a suitable choice of that curve; i.e. to deterniine whether it shonid be algobraical, cxponential, trigononictrioal, etc. There is indeed very little to justify the reailiness with which in practice a parabola of one or another order is sclccted to describe the results of Observation. K. Pkarson 17 A litlK' coiisidoratidii will fri'(|iii'iitly Icad Ui llie selection of a curve witli as few or oveii fewer coiistants giviiig a f';ir bcttcr fit. Thiole's observatioiis are giveu in § 6. Wc shall attenipt to fit a parabola- series to Thiuic's trapezoidal polygon bctween a'=(J and « = 20, i.e. we shall take 2^=14. The origiu tor moiiunit.s was takun at .«=13, and the siiceessivc niorrients in„ixi, »ififJ'2, vi„fi.,', )ii„Hi, »'o(".i' '»'id )ii„/J-,! caiculati'd tor the System of trapezia fniia the couceutiated ^'.s hy the lorniuhe /ii' = l'i', /i/ = ('/ + j'./c' + ^^c\ fJ-2 = '■■/ + ic", /X5' = v^' + ^ v^c- + i j// c\ fJ-s =W + ^ l'i' C\ Me = ''/ + 5 W C- + l'o C' + ^V C'^, where c is the base element, or in our case unity, and These formula; are deduccd in my memoir on " Ökew Variation in Homo- geneous Material," Fhü. Trans. Vol. ISG, A, p. 349*. See also § 10 above. Theu\s = ^ was caleulated, and the following value.s obtained : X, = - •162,857, X, = -114,748, X3 = - -033,778, X4 = -030,712, X^ = - -010,204, X„ = -012,141. In addition we have y„ = 3.5-7l43, Mean a,'= 11-86. From these values the es were caleulated and the following series of parabolas obtained, x being measiired from the mid-range : (i) y = 3.5-7143 {1 - '488,571 (x/l)}, (ii) y = 35-7143 [1-81!),(;94 - -488,571 (*•//) - 2-45;),082 {.i-;i.)-], (iii) y = 35-7143 {1-819,694 - 2-166,885 (a-ß) - 2-459,082 (.■r/0=+ 2-797,191 (a;/Z>'), (iv) y = 35-7143 {2-086,513 - 2-166,885 (x/l) - 5-127,275 (^vßf + 2-797,191 {.v/iy + 3-112,892 (*-/01, ■ (v) ?/ = 35-7143 {2-086.513 - 4-026,295 (a-/0- 5-127,275 (.r//f+ 11 -474,432 («/O'' + 3-112,892 {xjiy - 7-809,518 {xßf}, (vi) (/ = 35-7143 12041,057 -4-026,295 (.-•//) -4-172,701 (,-/0-+ 11-474,432 («/O» + -249,l70(,,7'/y - 7-809,518 {.vßf + 2-100,062 (xß^'}. * These are not the proper formultc if we coiisidered Thiele's observations as the areas of a frequency curve, but what we are here doing is to fit a series of curves as closely as possible to a polygonal area. Biometrika 11 i 18 Oll thc Sj/fifematic Fitting of Curves The ordinates for these curves corresponding to the original observations, i.e. + a-y'/=0, 1/7, 2/7, 3/7, 4/7, 5/7, 6/7, 1, were a.scertained, aiid are given in the accompanying Table. The curves are plotled in Fig. ü. Ordinales of Thiele s üb.iei'vations. Parabolas Äctual Numbers Skew Freqiiency Ist 2Dd 3rd 4th 5th 6th 6 •06 18-3 -40-3 - -3 251 -10-3 207 7 3 1-89 20-8 -14-5 -3-0 -3-4 5^8 -7-8 8 7 13-55 23-25 7-7 13 -8-8 4^8 14-4 9 35 42-26 25-7 26-3 107 1-0 39 50-5 10 101 77-29 28-2 41-4 236 19-3 112 797 11 89 97-49 30-7 52-8 38-0 40^5 28^25 929 12 94 93-54 33-2 60-7 52-4 601 51-5 892 IS 70 72-76 35-7 650 65 745 74-5 729 U 48 47-94 38-2 65^7 74 81 ^6 90-2 50-5 15 SO 27-60 40-7 62-8 77-6 80^1 92-4 28-8 16 15 14-21 43-2 56-3 74-2 69-9 780 12-9 n 4 6-46 45^7 46-3 61-9 522 493 5-1 18 5 2-88 48^2 326 3905 28-9 153 3-9 19 1 1-16" 50-7 154 3-9 34 -5-8 3-7 20 •44 53-2 -5^4 -45-3 -199 155 -5-1 Taking the (Jth parabola as the best let us compare the results found froni it with those obtaiued from the skew curve. Let A, be the difference froui Observation ia the latter case, A., iu the fonner. \Ve find \ A., - -1 -20-7 + 1-1 + 10-8 - 6-6 - 7-4 - 7-S - 15-5 + 23-7 4 21-3 - 8-5 - 3-9 + ö + 4-8 - 2-8 - 2-9 \ A, - 1-9 - 4-5 + 2-4 + 1-2 + -8 + 2-1 - 2-5 — 1-1 + 2-1 + 11 — 2 - 2-7 - -4 + 5-1 New whether we ineasure the g4 15 16 17\ 18 ♦0 -1 *2 ♦ 3 +4 \^^ .5 -\ '"V-- 6 \ 20 6 5 / / _Q POLYGON OF THIRD PARABOLA ^ OßSERVED FREQUENCIES \ f ; •3 STRAIGHT LINE FOURTH PARABOLA 2^ / . _ SECQND PARABOLA FIFTH PARABOLA \ \ SIXTH PARABOLA Flu. 9. Exauiple of fittiiiK parabolas (Thiele's frequeucy observations). with, perhaps, a qnarter the number of constants. Heuce the sort of statenient so frequently heard, — " Yes, of course, more constants make better fits," — is oiily a half truth, and the manner in which engineers, physicists and actuaries so readily use parabolic curves is open to consideral)le criticisin. There are offen considerations, hing outside the actual data, which suffice to iudicate that trigonometrical, exponential or other types of curves will give better results than parabolas. A parabt)la which passes even through all the observations ma}' indoed 3—2 20 Uli tlic Syatematic Fi(ting of Curces be a inost iindesirable reprcsuntatiuu of thc facts, für it has twisted and curled to account for error as well as to give the general sweep of the observations. (l-i) Illustnition VII. As a secnnd illustration of parabola-fitting I will take froiu the Italian inarriage statistics of Perozzo thc modal ages of bridegrooms of brides of giveii ages. With some simple interpolations I have determined these from Perozzo's tables* as approximately the following: Age of Probable Age Age of Probable Age Bride of Groom Bride of Groom 15-5 31-5 33 16-5 25-2 32-5 33-5 17-5 25-4 33-5 34 18-5 25-5 34-5 34-5 19-5 25-5 35-5 36 20-5 2Ö-5 36-5 37 21-5 25-75 37-5 38 5 22-5 26 38-5 39-5 23-5 26 39-5 41-5 24-5 26-8 40-5 41-5 25-5 27 41-5 42-5 26-5 27-5 42-5 43-5 27-5 28 43-5 43-5 28-5 29 44-5 43-5 29-5 30 45 5 43-0 30-5 32 New let US take 43'5 as the origin of age for thc mau, and 30'ö as that for the woman ; then if i/ be the iiian's age aud .v the wouiau's, we have thu following series of poiuts : x= 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, y=ll-.3, lü-ö, 10, 9-.5, 9, 7-5, 6-5, 5, 4. 2, •_', 1, 0, 0, 0, 0, a-= -1, -2,-3,-4,-5, -f), -7, -8, -9,-10,-11,-12,-13,-14,-15, ?/= 13-5, 14-5, l.rö, IG, lü-."), Ki 7, I7ö, 17ö, 17-75, 18, 18, 18, 181. 18-3, lS-5. The ränge is thus 2^ = 30, and conuuntrating and then using the eorrective forinulaj for trapezoidal polygons given in § 10 we easily find y„ = 10-78G7, \, = _ •348,5000, \, = 173,6922, X,= -299.9504, X^ = - -137,7881. X;, = - ■19.s,1573, X,. = -1 23,0257. The e's were then determined from the X's by the forraula; of § 12 aud we have for the series of parabohus, if / = 15 yeai-s : * Reale Accademia äei Lineti. Aiiuo ccLXXix, 188'2. Nuove Applicazioni del Calcolo delle ProbabilitK. K. Pkarson 21 ] st parabola : y = 10-7.S(>7 \\ - 1-0455 f j]! . 2n(l parabola : y= lOwMdT |l-lL>.-,,lG3.5- 10455 ^^j - -375,4905 f^Vl . 3rtl panibola : y = 10-7867 |l-125,1635 - 1-332,7459 {^^ - -375,4905 (^^y+ •478,7431 (''^'j 4th paiabola : y = 10-78G7 |l-159,1392 - 1-332,7459 (1) --715,2475 (^ xy „„.. , fx + -478,7431 (^1 +-396,3832, öth parabola : y = 10-7867 ] 1159,1392 - 1-428,4973 (j) - -715,2475 (j + •925,5752(1)' + -396,3832 (^V- -402,1555(^1 6th parabola: y = 10-7867 1-108,0239 - 1-428,4973 (|j + -358,1736 (j\ + -925,5752 (j - 2-823,8816 f-"")'- -402,1555 fljV 2-361,5275 ['*' The (inlinates of these six curves were then calculated for thc 31 values of x, and the curves theinselves with the observations plotted in Fig. 10. We have again an instructive graphical representation of the closer and closer approach of a series of parabolas to a fairly smooth System of observations. It will be seen that the parabola of the 4th order gives quite an excellent representation of the observations, better indeed tlian the 6th parabola which has tuo niany points of infle.xion to dispose of (15) With this illustration I elose my discussion for the present of curve- fitting. I have emleavoured to shovv : (1) that the niethod of nioiiients inust theoretieally give good fits ; (2) that it provides a systcmatic inuthod ol litting a great variety of curves ; (3) that it is over and over again available when the method of least Squares fails, or can only be applied with excessive labour; 22 On the Si/sfematic Fitting of Curves ' V • »9 \ 31 6 \ » 9 i 4 ■ '^-X , 19 4 ""^^^ii:ii=_i-A *- -*• IB "*. \ °"V.>^?^fe^x 's. 3 / ■ *.,«^ ' V^ IT \ ■• ^^ 16 \ ■■ >Vi. \ ■ '^ \ '•• A ■ IS \ ■ '.% \ . ^\ 14 \ ■' A \ ■ ^x \ H IS \ % \ V \k IJ Vv^ , 11 \k^ \ N^\-. 10 \ vp. ■ « \v L ^ B I 5 i A ^\"''- \' 3 Vv \ V\ . s' 'H-;>. \ / \^'*A i-^ 5 14 t3 13 1) 10 9 e I65«3J10153» 1 s , POLVOON Of TMIBO PiRAOOLA •■ > } ^ OBsenveo frequcncies DAla>DlM & STRAIQHT UNE .._ FOURTH ....... 8EC0N0 PARAQOLA FIFTM PARABOLA .._ SIXTH PARABOLA 3 Flu. 10. Exumple of fitting parubolas (Itiiliun iimrriage statistics). K. Pkarson 23 (4) that in all cases much rleponds on the method by which the successivc momcnts are calculated, and thafc fn-quency observations and physical measiiro- nicnts reqiiire to sonie extent dirt'crent processes ; (5) that with a good nioment fornuda the method of nionients will give for practica! purposes as good fits as the method of least Squares when both are applicable ; (G) that für fitting parabulas the method of inoments has theoretieally the same basis as that of least Squares, but in application it is much easier; and lastly : (7) that far more really depends for goodness of fit on the eqnatioii to the curve selected than on the nnmber of its constants, and that the multiplication of constants to improve fit is not only theoretieally undesirable, but does not necessarily lead to the required residt. QUANTITATIVE STUDY OF THE EFFECT OF ENVIRON- MENT UPON THE FOUxMS OF NASSA OBSOLETA ANÜ NASSA TlUVITTATA FROM COLD SPRING HARBOR, LONG ISLAND. Bv ABIGAIL CAMr DIMON. (1) Tntrodiictori/. The aims of this paper are to niake a i]uantitative inqiiiry iuto the effect of" diverse eiiviiontiicntal couditions upoii the form of two gastropod species, Nassa obsoleta aiid Nassa trivittata, from Cold Spring Harbor, Long Island, and also to record the characteristics of their shells and thns to determine the " place inode " for these shells in that locality. The chaiacters selcctcd for measurenient were those described in systematic works as distinguishiug the species as far as those characteristics could be easily dotermined quantitatively. The work was done under the general direction of Dr Charles B. Davenport. (2) General Description of Nassa, its ränge and kabits. Xassa. is a genus of piosobranch gastropods coiitaining many species distributed over the whole World, chiefly in shallow water. The individuals are usnally small, with an ovate shcll and a large foot, which is notched behind and carries a homy opercuhim. The two species, Na.ssa obsoleta, Say (Ili/anassa obsoleta, Stiinpson ; Biiccinum obsoletum, Gould), and Nassa tnvittata, Say (Tntia trivittata, Adams; Biiccinum trivittatum, Gould), found commonly at Cold Spring Harbor, are Americtm forms, and with the less common Nassa vibex constitute the oniy recognised littoral species of the genus found on the middle Atlantic coast of tho United States. Both species ränge froni the Gulf of St Lawrence tn I''liiii(ia. \'irrill ("!')) reports N. trivittata as abuiidant at Casco Bay, Maine, and in Vineyard Sound and Buzzard's Bay, and as common along Long Island Sound ; whoreas N. uhsoleta. he reports as very abundant soutli of Cape Cod and more ioral lurtln r iHntli. N. trivittata, thercforc, reaches its maxinunn inimbcrs further nurlh tlian N. obsoleta. Geologically, N. trivittata is oider than N. obsoleta, having been found in the Miocene of Maryland. Virginia and South Carolina, while N. obsoleta has not been reported fiiuii t'urther back th.in the Piiocene. A. C. DiMON 25 ( '(iinpiiriiig in detail tlie liabitfits ul' tlio two spccies, it is sccii that N. obsoleta livc's in lari^c nuniber.s on tiats and .sliorcs Icf't bare ])y the tidi' dui'ing part of thc day, aiul is not f'ound at auy considoiablc depth ; whilo N. trivittata is found in sonio places at a dupth of forty fatlionis, und is not f'onnd abovc low wator line. Verrill ('7.S) statcs that about Vineyaiil Sound jY. ubsoleta ooours in bays and Sounds on sandy and niuddy shorcs and bottonis, and on submorged wood-woik, such as the piles of wharvcs, but that it does not occur on rocky shores and bottoms. N. obsoleta was found in brackish water on sand, nuid, oyster-beds, eel-grass, and subnierged wood-work ; but it was not found in the open ocean. N. trivittata was found by Verrill in all the habitats of N. ohsoleta except the muddy shoi-es of the bays and sounds aud the muddy shores and bottonis, oyster- beds and eel-grass in the brackish water ; it was found in the open ocean on shores and bottoms. With these observations my own expcricnce with the distribution of the two moüusca at Cold Spring Harbor and elscwhere is in complete accord. The breeding season of N. ohsoleta is given by Mead ('98) for Woods Hol! as the latter part of April. In July and August when nearly all niy collccting was done there would often be found in pools left abovo low water mark large numbers of very small individuals which were evideutly the brood of that year. The older snails, however, could not be separated into broods of different ages on the basis either of size or the number of whorls, so that eithcr the growth aftcr the first year is extremely slow or eise the snails do not live until the third sumnier. (3) Localities froiii which tite sJiells were collected. At Cold Spring Harbor we have, within a small district, several localities in which Nassa may be found. The Harbor is a brauch of Long Island Sound, five miles long by one mile wide. It is divided into an inner and an outer harbor by a sand-spit that extends nearly across it at about half a mile from its head. Tlie inner harbor is fed by a stream which makes its wator decidedly brackish, espocially at the surface. For ncar the niouth of the creek, at the surface, the den.sity may be as low as l'OOG whiie at thc bottom it is I'OIG. Undor other conditions of wind and tide the density will be about I'OIG throughout. The average height of tide is about 7'.5 feet in the inner harbor, and at low tide about half the surface is left bare, exposing flats of black mud on which Ulva grows in abundance. In the outer harbor the density of the water is from I'OIS to 1020. The bottom of the outer barbor consists of mud, oyster-beds or sand, with a good deal of eel-grass ; the shores are Sandy or muddy with a greater slope than those of the inner harbor, so that there are no extended mud-flats exposed at low tide. At the mouth of the harbor, on the east, thc shores are sandy and gravelly and have a considerable slope, and the Situation is far less sheltered than within the harbor. The A^assa measured were collected from the three localities marked 1, 2, and 3 on the map, Figuve 1. Those from 1 (Laboratory Dock) were gathered from the mud-Üats at low tide ; those from 2 (Laurelton Dock) were taken from the sandy beach at low tide or dredgcd from a few feet of water at the same locality ; and those from 3 (Lloyd Point) were collected from pools left on the sand beach Biometrika ii 4 26 Study of N. ohsoleta and N. trtvittata at low tidc. It was not possible to find N. trivittata in abundancc in all thrce localities, so this species was taken only fi"om 3, where the shells were washed up on the beach in considerable nunibers and easily collcctcd. Only one or two specimcns were di'edged l'roni locality 2, and no specimen has been found in the inner harbor. Fig. 1. Sketch Map of Cold Spring Harbor. (4) Qualitative comparison of the shells Jrom the different localities. If hauflfuls of N. ohsoleta from each of the thrce localities niarkcd on the map be compared a decided diffcrencc in sizc will bc at once noticcd. The shells from 3, the most exposed locality, are much the largcst. On the other band they are niiich niore nunierous in the inner harbor. In comparing N. trivittata witli N. ohsoleta it is seen that the shells of N. trivittata are not covercd with algae and are not erodcd, while the shells of N. ohsoleta are covered by algae and much eroded at tip, probably in consequence. Also lY. trivittata is almost white, whereas N. ohsoleta is blackish jnirplc, its apical angle is morc acute, and its shcll is smaller. (5) Method of measuHng. The characteristics of the shells of which it was sought to get a quantitative expression were size, shape, number of whorls, color, and roughness of surface. To ensure accuracy the dinieiisions of the shell and the angle at its apex were all measured two or three tinies. On accoiint of the length of the process or the great effect of a previous reading on the judgment A. C. DiMON 27 the determinatioiis of the nuiiibor of whorls, oolor, and roughness wcrc made oiily once, so thcse dutermiiiations ai-c less accurate. To measuro length a niicrometor gauge reading to hundredths was used. In Figiire 2 a, the line AD represents the diroction in which length was measured, B the direction for diameter, and C for greatest length of aperture. The sliape of the shell was given by the angle at the apex and the ratios of the diameter to the length and of the aperture to the length. The ratios were calculated fnmi the nieasurements ; the angle was measured directly by nieans of a bevel protractor. The erosion of the apex Fio. 2 a. Fia. 2 B. disturbed the measurement of the angle in the case of N. obsoleta, and it was finally decided to takc an angle the direction of each side of which would be a compromise between the directions of lines drawn between the center points of successive whorls. D, Figure 2 a, shows such a compromise between lines 1, 2, 3, on one side, aud 4, 5, on the other, and represents what I have called the apical angle. Even after the niore eroded shells had been discarded, the bluntness of the apex often affected the angle, the geueral tendency being to read it too large, as showu in Figure 2 B. The niean angles of shells fi-oin the different localities are therefore to be compared only with caution. 4—2 28 Stiidi/ of xV. ohsoleta and X. triiittata The nuinber of whorls was counted directly. Color was ineasiired by the color top in the region E, Figure 2 A ; the results are, however, rather rough. The depth of the suture between the whorls was chosen as an approximate lueaKure of ronghness. It was measured between the last two whorls, near F, Figure 2 a, by laying the shell so that it rested on its aperture and making a camera drawing magnified eight or niue tiuies of the part of the line it w;\s desired to measure. Then the depth of the depression from a tangent drawn to the adjoining elevations was measured on the drawing, and this distauce was dividod by the magnific^ition of the figure. The diniension thus obtained depeuds, however, not solely upon the roughncss of the shell but also upon the convexity of the individual whorls. Figure 3 shows some types of outlines obtained in the nianner described ; h rcpresents au e.xtreme case of a sniooth shell with a deep suture. Fig. 3. Camera drawings illustrating tlie depth of suture. S = Shoulder. (6) Table üf constants of the curves. In the foliuwing table some of the results of the quantitative detenninations are given. 1, 2, and 3 represent lots of N. obsoleta from the localities referred to ; 3a is the lot ot N. trivittuta from Lloyd Point. Discussion of constants. Means. Setting aside for the present N. trivittata, a comparison of the means of JV. obsoleta from the tliree localities brings out certain points. The sheils from 3, the mouth of the harbor, are much longor than the others; those from 1, the inner basin, are the most globose, and have the largest aperture in proportion to thoir size, and the fewest whorls. In dopth of suture the sheils from the mouth of the harbor are e.Ktreme. The probable error of the color measurement is so great that uo stress can be laid on the slight difforences recorded. As to the apical angle, the effect of erosion at the apex upon this reading is, as already stnted, .so considerable that lot 1 (the most eroded) is not strictly companible with the other lots. If we coiTelate these dififerences with the environmental differences noted above for the three localities, we find the iudividuals living in less dense water — in the inner harbor — are smaller, more globose, with a larger aperture in proportion A. C DiMON 29 to their length, and a shallower suture. Thoiigh the series is not in eacli case a rcffular one, with the shells from 1 and 8 at the cxtremities aud those froni 2 intennediate, yct the gencral relationship holds. The tendency of the shells früiii 2 toward an t'xtivine position i.s not necessarily inconsistent. Cooke ('95) TABLE I. Quantitative dafit. Table of Constants of the Curves. Mean Probable Error of ,1/ Observed Mode Theoretioal Mode Staudard Deviation Probable Error of o- Coefiicient of ■Variability Type Skewness Length : 1 13-92 mm. + -0336 mm. 13-50mm. 13-79 mm. 1-120 mm. + -0238 mm. 7-86% IV + -1206 3 12-7Ö „ + •0372 „ 13-00 „ 12-76 „ 1-232 „ + -0263 „ 9-10 % IV - -0051 S 17-14 „ + -0Ü64 „ 17-00 „ 16-46 „ 1-890 „ + •0469 „ 11-03% I + -3590 Sa 11-96 „ ±•0814 „ 12-00 „ 12-30 „ 2-590 „ + -0575 „ 21-66 % I --1328 Diameter Length ' 1 59-74% + -109°/„ 57-50% 59-09 % 3-614 7„ + -0773 7„ 6-05 % I + -1796 o 58-30 7„ + -103 °/ 58-50 7„ 58-37 °/o 3-416 °/„ ± -0760 % 5-86 °U IV - -0216 S 58-71 7„ + -099 7„ 59-00 % 55-56 % 2-825 7„ + -0714% 4-82 7„ IV + •0608 Sa 52-14 % ± -118 70 51-50 7„ 51-30% 3-740 % ± -0835 % 7-17 % IV + -2238 Aperture 64-12 7, + -107 7„ 64-50 °/„ 64-25 °/, 3-522 % + •0754% 5-49 % IV - -0354 Length ' 1 Q 62-32 7„ + -080 °/ 62-50 °/„ 62-63 °/„ 2-646 °/„ ± -0565 7„ 4-25 % IV --1178 s 62-71 7„ + -092 7'^ 63-00 ",/ 62-84 7 _ 2-626 °/„ + -0652 °/„ 4-19% IV - -0481 Sa 51-24 7, ± -oyG 7„ 49-00 °/„ 50-87 % 2-727 °/, ± -0608 °/„ 5-32 % IV + -1350 Apical angle : 1 53-44° + ■176° 57-00° 53-93' 5-832° + -1241° 10-91% I - -0844 2 55-46° + •151° 57-00° 55-79° 5-046° + -1071° 9^10 °/„ I - -0659 3 53-09° + -200° 53-00° 53-30° 5-702" + -1416° 10-74 % I - -0361 Sa 47-48' ±•129° 48-00° 47-17° 4-102° ± -0910° 8-64 % I + -0766 Numberof whorls : 1 4-88 + -017 5-00 4-63 -560 + -0120 11-48 °/„ I + -3790 2 5-49 + -015 5-25 5-43 -496 + -0106 9-03 7„ IV + -1161 3 5-34 + -014 5-38 5-36 -404 + -0102 ''■^~ 7o I - -0379 Sa 7-01 ±•019 6-87 7-16 -595 ±-0113 8-49 °/; IV - -2480 P. c. of Red in color : 1 10-44 °/„ + -063 °/„ 10-00 % — 2-063 % ±•0447% 19-76 % — — '>> 10-09 "/„ + -076 % 9-50 °/„ — 2-546 °/ + -0542 "/, 25-17 7o — — 3 10-51 % ± '065 °/„ 10-00 % 9-20 °/„ 1-860 % ±-0462% l--29% I + -7018 P.c.of Black in color : 1 2 o 88-44 % 88-38 °/, 88-11 7„ ±•072% + -0S4 °/, ±-083% 90-00 7„ 89-50 °/„ 89-00 °/„ 88-92 7„ 2-336 % 2-798 % 2-352 % + -0506 °/„ + -0596 °/, ±-0585% 2-64 °/„ 3-17 % 2-12% I - -3465 Depth of sutiire : 1 -»»w *-w 24-00"""- 100 ««TO o"-^'^ *-»-S- 31-40% I + -3895 2 19-83 „ + -194 „ 18-50 „ 17-61 „ 6-423 „ + -1405 „ 32-39 % IV + -3458 3 33-35 „ + -418 „ 28-00 „ 27-20 „ 11-890 „ + -2956 „ 35-65 °U I + -5177 Sa 42-15 „ ±•299 „ 43^00 „ 41-75 „ 9-530 „ ±-2117 „ 22-61 % IV + -0420 30 Study «. lotal frequency, 458. 1 ■ Chiss ... j ^ '■:410 2 420 3 430 4 44-0 2 5 45 1 6 460 7 47-0 8 48-0 9 490 10 .50-0 11 510 12 520 53-0 14 54-0 15 55 i6 560 17 57-0 580 i9 59-0 20 600 61-0 Frequency 1 7 15 27 72 71 63 63 51 35 21 14 7 2 1 4 1 Angle: Locality 1. Class ... Frequency y ^ Total frequency, 502. j 35 2 37 3 3 39 3 41 8 5 43 12 6 45 24 7 47 8 49 9 51 53 11 55 12 57 i5 59 «1 25 76 65 17 67 i5 69 19 71 1 39 48 56 65 65 73 43 30 u 8 4 1 Angle: Locality 2. Class ... I Frequency Angle: Locality 3. Class ... I Frequency Angle: Locality Sa Class ... I Frequency 1 2. Total frequency 505 1 .' J, 5 6 7 S 'J lii /; ;.' 13 14 41 43 45 47 49 51 53 55 57 59 61 63 65 67 3 .') 1 n 21 .•il .-.6 61 73 77 73 50 33 9 3 5. Total frequency, 369. / 39 2 41 6 5 43 15 4 45 21 5 47 S 49 j 51 53 9 55 10 57 11 59 12 61 63 9 14 65 5 25 67 2 i6 69 1 29 38 1 35 58 52 40 30 26 / 3a. Total fr •UIUIIC-V. n-2. 1 38 40 42 44 46 48 50 r,-2 ."■1 .-..; /; 5 19 35 73 77 94 77 36 31 10 5 1 W'/ki/'/s: Lora/iti/ I Ckssi ... I Froquinic-y l\'/ior/s: Loctdity 2. Clas« ... I Froi[iuMR'y \\7ior/.i: Lorulitt/ S. Class ...(I ^ l 4-l:: Frequoncy ' 1 A. C. Dlmon Total frequency, 493. 37 1 4-00 2 4-25 S 4-50 4-7r) 5 5-00 6 5-25 7 5-50 8 9 6 00 10 ()-25 24 75 71 81 110 51 29 23 23 6 1 'r< tal f •cqucncy, 494. 1 4-25 2 4-50 3 4-75 11 5-00 5 5-25 5-50 5-75 94 S 6-00 60 9 6-25 15 10 0-50 4 11 0-75 3 1 4 62 138 102 Total frequency, 357. 2 .? 4-38 4-63 7 ' 7 4 4-88 65 5-13 61 6 5-38 102 7 563 55 48 6-13 9 70 6-38 Wliorls: Locdllli/ .Ui. Total fVequeiicy, 439. Class ... 1 4-88 2 513 3 5-38 5-63 5 5-88 6" 6- 13 7 6-38 S 6-63 9 ß-88 10 713 11 7-38 12 7-63 7-88 U 8-13 io 8'38 iß 8-63 Frequoncy 2 1 2 6 20 13 35 37 86 84 68 42 36 4 2 1 Redness : Loi'id Vy 7. Total frequency, 484. Class ... 1 1 7 -2 8 S ; .4 9 10 .5 11 0' 12 13 ,S' 14 y 15 12 10 16 2 ii 17 1 12 18 3 i.5 19 2 u 21) i5 21 2 Frequency 11 53 84 150 81 49 23 11 Riducas: Londi ty 2. Total frequency, 502. Class ... 1 / 6-5 6 2 S 7-5 8-5 9-5 10-5 11-5 12-5 13-5 14-5 10 15-5 11 16-5 17-5 18-5 i^ 19-5 15 20-5 IG 21-5 Frequency 87 1 94 118 73 41 24 19 16 3 8 4 3 4 2 Redness: Locality 8. Total frequency, 368. Class ... 1 1 6 2 7 3 8 -4 9 5 10 6 11 7 12 8 13 9 14 10 15 16 ig 17 Frequency 2 5 40 47 113 \ 68 ; 44 30 9 8 2 38 :Stu(hi qf N. obsoleta and JV. trivittata Blachnets: Locality 1. ToUl frcquoncy, 484. Class ... I Freqiiency Blachneas: L Class ... I Frequency 1 2 77 78 S 79 4 80 5 6 81 82 7 83 8 9 \ 10 84 85 86 11 87 12 13 88 89 U 90 15 91 16 92 17 93 l 1 1 1 4 5 3 8 1 25 j 34 49 76 97 106 55 15 3 ocalit y2. ToUl frequency, 502. 1 ! 5 76-5 77-5 5 78-5 -* 79-5 5 6 80-5 81-5 7 82-5 83-5 9 84-5 10 85-5 11 12 13 I i.J 86-5 87-5 88-5 89-5 15 90-5 16 1 17 91 -5 '925 2 1 2 4 6 1 7 7 11 16 15 33 65 91 1 99 1 73 61 9 Class ... I Frequency ocalit^ / •*• Total frequency, 368. 1 81 2 2 *^i! S S3 84 5 85 6 86 7 87 88 9 89 10 90 91 12 92 75 93 94 5 6 16 28 27 41 60 73 65 28 14 2 1 Class Frequency tture: Lc catit^ y 1. Total frequency. 456. ■il 2 12 3 15 4 18 5 21 6 24 27 8 30 9 33 10 36 ii 39 12 42 i5 45 14 48 15 51 ,y 6 15 32 48 60 65 64 62 41 28 15 6 8 2 4 pi/j 0/ Suh tre: £oca^ % 2 T otiil frequency 498 ass ... -1 1 8 2 11 3 14 4 17 5 20 6 23 26 29 32 10 35 11 38 12 41 J5 44 47 15 50 i6 53 17 56 59 requency 6 42 76 102 102 7s h; \-< 1 1 9 4 .) 1 11 n 1 Cla-ss ... ■ Frequency LocaliUi 3 'I ■otal frequency , 368 1 8 13 18 23 28 33 38 43 9 48 10 53 11 58 63 68 73 15 78 16 83 1 5 27 67 --I 47 34 20 11 14 3 3 1 1 f Suture: Locality Sa. Total frequency, 461. Class ... 1 1 ! 2 18 23 L'S 5 ' .- ' '.• ! 7 ■Vi 3S 13 48 8 53 9 58 10 63 11 68 12 73 IS 78 Frequency 4 15 28 53 99 109 06 50 30 3 3 1 A. C. DiMON .39 APPENDIX B. Corixhttioii Tiihli:i. TIki olasscs arc lieliiu'il liy \\\v cI.lss iiuiulicrs of tlif Variation »tatistic». a. Correlation bctweon Icngth and niunbor of wliorls of sliells of X. olmilda. froui locality 3. Lwigth : Ninnbor i>f Whorls : Jlean = 5'0983. Mean = 5-8286. Standard Deviation = r868. Standard Deviation =r573. Correlation = '0040. Total frequency = 35ß. Numbcr of Whorls 1 3 S 4 5 6 / 8 9 10 1 _ _ 1 1 1 _ __ ) — — 2 6 6 4 1 — _ — 1 1 11 6 9 13 7 1 J, 1 2 1 13 12 14 9 12 1 1 5 — 1 2 20 13 23 9 8 2 ß — — 2 9 12 13 4 10 1 1 — 3 — 8 6 15 5 4 3 — 8 — — 1 — 5 14 7 2 1 9 — — — 1 — 6 3 3 — — 10 ' — ■ — — — — — — 1 b. Correlation between length and number of whorls of sliells of N. tririttatn from locality 3. Length : Number of Whorls : Mean = 7-879. ' Mean = 9-5548. Staudard Deviation = 2-594. Standard Deviation = 2-373. Correlation -— 4178. Total frequency = 4.38. Number of Whorls 1 '2 ,1 o k 5 6 / 8 9 11) 11 12 13 u ir, 16 1 — — — 1 5 1 1 2 2 — 1 2 — — — — — — — — 3 _ 1 5 3 1 _ _ _ 4 — — — 1 — 1 5 1 8 1 — — — — 5 — — — 1 — 1 4 10 13 5 2 2 — — x: — — — 1 3 1 1 10 14 12 4 2 2 — to 1 — 1 _ 2 — 2 5 13 14 13 1 1 — •^-^ s 1 — — — 1 2 2 4 17 24 12 6 3 1 — 1 — — — _ 5 4 10 13 15 9 8 — — 10 — — — 2 2 — 3 — 5 7 12 11 1 2 — 11 — — — — 2 6 — 2 4 4 7 12 — 12 — — — — 3 3 2 — — 3 4 2 5 1 1 1 13 — — — — — _ — — 1 1 1 1 3 1 n — — — — — — — — — — — 1 — — 15 — — — — — — — — — - — — 1 — — — 40 Study of N. obsoleta and N. tririttata c. Correlation botwecn leiigtli aml immlior of wIkipIs of sliells of iV. obsoleta froni locality 1. Length : Nuraber of Whorls : Main = 7-856. Meaii = löO'). Standard Üeviivtiou = 2'21öl. Standard Deviation = 2"16-12. Correlation = -2860. Total frcquency = 493. Niimber of Wliorls 1 ' .' .-v -'/ ,'i l! ,s .0 Kl / _ _ 1 _ 1 f? — — — — — — — — 3 3 1 — h 5 3 1 4 1 1 1 — , 3 3 15 10 10 8 1 — 3 1 .s 6 8 13 8 10 14 5 1 2 1 ü / 6 14 in 17 1!) 6 3 2 5 1 " S 2 11 11 17 i:i 11 4 6 1 1 9 — 8 7 11 11) 10 11 6 8 — 10 — 5 9 5 k; 11) 3 1 1 1 11 — 3 2 4 10 4 4 2 1 2 12 . — 2 1 2 1 1 2 IS — — 1 2 — 3 — 1 14 — — 2 — 1 1 1 — — 15 — — — — 1 — — 1 — d. Correlation between lengtb and ratio of diameter to lengtb of shells of iV. obsoleta from locality 3. Dianieter Length : Length Mean = 5-1 36. Mcan = 8-()89. Stjuidard Deviation = 1 '902. Standard Deviation = 2'797. Correlation= - •2641. Total frequency = 307. Index 7 2 3 J, ■' G 7 S m ;/ 12 13 U 15 IG 17 1 _ _ _ _ _ _. 2 1 2 2 — — — — 1 1 1 2 6 1 3 1 3 3 1 2 1 — 2 2 5 6 6 5 4 9 5 2 1 tr, k — — — 2 3 7 6 7 7 9 13 8 2 2 2 J 5 — 3 1 5 1 8 6 13 16 6 7 6 3 2 1 6 — 2 — — 2 2 10 7 13 10 3 2 2 1 1 7 — — — 1 3 10 8 5 8 6 4 1 — 8 — 1 3 10 6 4 3 1 1 ■_' 9 10 — ■ 1 — 1 2 1 2 4 3 — — — — — A. ('. Dl.MON 41 <'. ( 'on-clutiun lictwcci] lciiu;tli and ratin i>r (liaiiictor to length nf sliclln of jV. tricittala froiii locality 3. Diameter Leiigth : Loiigtli Moan = 7-9(;29. ^[can = 6-3ir)8. Standard I)cviation = 2-() 12. Standard Deviation = 1-870. (.'orrelati(in= - '"lOS. Total froqnoiK'y = 456. Indc.K 1 3 ^ r, ß / S 9 10 '' 12 j 1 ^ _ _ 1 _ 1 _ o .')' — — — — — — — — 6 1 5 5 4 2 -^ — — 1 — 1 7 3 3 2 — 5 2 2 11 15 8 1 — — f ('■ , — 2 12 10 15 8 2 — — ÖD a 0) 7 — — 11 18 15 6 4 — — — 3 4 5 6 7 8 9 10 11 12 1 2 2 4 10 4 1 1 1 — 2 16 16 17 :« 1 17 — 3 1 2 ■2(\ 35 ? 1 — 2 18 42 19 10 3 1 5 13 33 16 7 1 3 7 2 17 4 6 13 1 8 2 3 1 2 1 3 3 1 1 3 1 A 0. DiMON 43 i. CoiTclation lictwccn nuiiiber of whoi-Is ;uid aiiioal aiigic of .sliclls of X. ohsnlcta from locality 3. Numlicr (if W'lioi'ls : Anglo : jrcaii = h-KVl. Mo.iu = 8-0784. Standard Deviation^ r()15. Standard Deviation = 2'85i). Correlati<)ri = -O077. Total frcquL'ncy = 35". Angle ^ ^ 1 > s k !t 6' 7 .9 9 111 1! 1^ ;.)■ u ir, /(-• 1 — — 1 — 1 — — 2 2 1 1 1 — 1 — — — s 1 2 1 1 _ _ _ _ J, — — 3 6 1 9 7 12 8 5 3 6 3 — 2 1 1 2 8 7 3 15 4 6 2 5 3 3 — 1 ti 1 2 5 10 11 10 11 13 14 11 6 6 1 1 — 1 — 1 3 2 3 4 3 8 10 5 7 7 1 1 — s — — 2 1 2 3 5 6 11 7 9 2 — — '.) — — — 1 1 3 1 — 2 1 — — — — 10 1 - — — — — 1 — — — — — — ON THE AMBIGUITY OF MENDEL'S CATEGOPJES. r.Y W. F. 11. WELDON, F.R.S. I. Introdudorij. In the early part of this year I piiblished an account ol' tln' results obtained by Mendel in bis experiments witb certain races of cross-bred petis, and of the subse- queut attempts to show tbat results similar to bis werc obtainablc in othcr cases*. I See no reason to niodify the Statements I then made. Since my articlc appeared tlic Royal Society has published a Report by Mr Bateson and Miss E. R. Saundersf, in which a large number of experiments in cross brceding are dcscribcd; Menders results are re-stated, and most of the observa- tions rccorded are hold to be in substantial agreemcut with Mendels laws. In this Report, and in a separate essay by Mr Bateson |, an altogether new interpreta- tion of Mendels results is suggested, and an issue of considerable importance is raised. Mendel says Ihat the races of peas, uscd in his experiments, were observed during several generatious, and were found to iliffer constantlyin certain characters, for exaniplc in the colour of tlieir cotyledons, which was grcen in sonie races, yellow in others. Now "green" and "yellow" are not quantitatively dcfinite terms; each includes a considerable ränge of recognisably different coloui-s, and every known race of peas produces sceds whose cotyledons vary in colour. All that can be in- ferred from Mendels statement, therefore, is, that the ränge of Variation in cotyledon colour, which some of his races exhibited, feil entirely within the ränge of colours called "green," while that of other races feil entirely within the ränge of coloure called "yellow." Mendel accounts for the bchaviour of hybrids between these races, as he duscribes them, by assuming that each hybrid bears gametes, male and female, of two kiuds and in equal numbers, "und diiss diese Keim- und Pollonzollen "ihrer inneren Beschaffenheit nach den einzelnen Formen entsprechen §." The • See Jiiomttrika, Vol. i. p. 228. t Rcpnrts to Ihe Kvolution Committee : I. Experiments undertaken by IT. Btitefon, F.R.S., aiid MUs E. It. Sauiiders, Uoyal Society, l;i02. t MrndiVs Prineiplex nf Ilereditij, a De/iiicf : by W. Biitesou. Cambridge University Press, 1902. § VerUandL d mitiirj'ursch. Verein» Jlruiiii, Jiil. iv. p. 2'J. W. i; l{. Wkldon 45 issuc raised by Mr Batuson aiiil Miss Saundcrs concenis Ihc iiieaiiiiig wliicli it, is ncccssary, in tlic light of kiiown l'acts, to givo to this exprcssioii. The rc>sult, described hy Mendel, niaybe illustrated by rcfcrenee Lu Uie Cullinving diagram : Let AD be a seale i)f' eolour, — a map nf a speetrum, or some similar lliiiig, — of" whieh tlie i'ange AB inelndes tlie variuus shades of green, the ränge CD thosc «f yellow. A race of peas has cotyiedons whose eolour varies, in individual casos, within the ränge AB; a second race has cotyledon.s which vary withiu the ränge CD. If these races be crossed, the hybrid cotyledons are said to fall within the colour-range CD; and the plants to which they give rise will, on Mendels hypo- thesis, produce gametes of two kinds, in ecpial numbers; those of the first kind, if paired, give rise to plants whose cotyledou eolour (and that of their descendants) lies within the ränge AB; those of the second kind, if paired, give rise to descendants whose cotyledon eolour lies within the ränge CD. That is to say, the descendants of such a cross fall iuto two group.s, each refer- able, so far as cotyledon eolour is concerned, to the same category as ono of the ancestors used in the original cross. Wc are quite unjustified, from the data given by Mendel or by any of those who follow him, in saying more than this. Thus the green-seeded descendants of such a cross are de.scribed as "green": they are not described, by Mendel or by aiiy of his foUowers, as being of the same shade of green as the particular member of a variable race used in the original cross. The information given is therefore compatible with either of sevei'al tlieories of the Constitution of those gametes in the hybrid frum which " recessive " individuals arise. For anything we are told to the contrary, the character transmitted by such gametes may be a blend of all tho kinds of green exhibitcd by their green-seeded ancestors in various proportions ; or different gametes may revert directly to the eolour of different individual ancestors; or finally all the recessive gametes may transmit the characters of the green-seeded ancestor which took part in the original cross, and of no others. The first two hypotheses involve the belief that the coniposition of gametes of either kind, whether "dominant" or "recessive," is affected by that of a whole series of ancestors. Such belief seemed (and still seems) to me a necessary con.sequeuce of the facts. Mr Bateson and Miss Saunders have however adopted the view that "the [niro dominant and the pure recessive members " of each generation are not merely like, but identical with the pure parents" (I.e. p. 12); and Mr Bateson denounces in no mcasured terms my attempt to regard the inconsistent results obtained by thosc who have repeated Mendel's experiraents as due to diflferences in the ancestry of the races used. Mr Bateson fully admits the variability of all races of peas ; and he admits that the ränge of colour-variation in many races is so great that they include colours of both the "green" and the "yellow" category. Nevertheless, he believes that many of these races behave, when crossed, like those described by Mendel. This 46 On MendeVs Categon'cs admitted variability of the races makes it possible for Mr Batcson to tcst his view expcrimcntally ; for if a "recessive" plant, resnlting froni a cross, is " not nicrcly liku but idontical with " one of the piire-bred plants used in the cross, then the peas on the recessive phmt must cxhibit exactly the same series of colour-varieties in their cotylcdons as thosc exhibited by the ptirc-brod plant which it resembles. Before Mr Bateson can justify the view he has put foiwaid, he must therefore not only find two races which obey Mcndel's laws when crossed, which he says he has done, but he must determine (1) The variability of cotyledon colour in each race ; (2) The mean cotyledon colour and its variability in each plant used in crossing ; (3) The inean cotyledon colour und its variability in each "recessive" and in each "dominant" plant descended from the cross. Until the result of such determinations is known, it is impossible to distiuguish between the rescmblance of a series of cross-bred plants to one of the anccstral races, and their rescmblance to an individual plant of that race ; so that Mr Batesou's contention cannot be supported by evidence. The confusion between rescmblance to a race and rcsemblance to an individual involved in Mr Batcson's treatmcnt of Mendels work is one of the many unfoitunate results which follow when Mendel's system of dividing a set of variable characters into two categories, and of using these categories as Statistical units, is carried too far. Unless the ränge of characters actnally includcd in each catcgory be constantly borne in mind, the degrec of rcsemblance between two individuals, implied by placing them in the same category, cannot be estimated ; and when, as constantly happens, the ränge of Variation in one of the alternative categories differs widely from that included in the othcr, the Mendeliaii System bccomes absolutely mis- leading without some explanation (nearly always withhcld) of the real limits implied by the terms used. Thus if two plants arc said to be glabrous, we know that thoy arc absolutely similar in so far that they possess no hairs ; if they are said to be hairy, we know that they both possess hairs, but one may, for anything we arc told to the contrary, have ten times as many hairs, per unit of surface-ai"ea, as the other. Again the tcndency to apply two categories, found suitable for a particular race, to other races of the same or allied species leads to very harmful results; for example the Classification of peas into thosc with green aud those with yellow cotyledons leads to a (piitc crroncous conception of the distribution of cotyledon colour in most existing races of peas, although it may have expressed the facts observed by Mendel in the races which he used. The Report of Mr Bateson and Miss Saunders contains many statcnients which would, I think, nevcr have been made if the authors had not been misled by the use of Mcndelian categories. It is impossible to realise the mcaning of evidence, brought forward to prove that particular hybrids behavc in the mauner described by Mendel, unless the meaning of the categories employed in eacii special case is W. F. R. Wkldon 47 clearly uiulerstciod. I tlu-reforo proposc, as opportuiiity offers, to describi^ the variability of such races as I caii obtain, ainong thosc which are said to obey Mendels laws, and to consider how far the Statements niade concerning them are affected by the euiployment of Mendel's very imperfect system of units. II. Lychnis diurna and Lijchnis vespertina. In the Report by Mr Bateson and Miss Saiindors the results of crossing normal L. vespertina and L. diurna with white and red-flovvered glabrous varieties are described ; it is said that the phenomena " follow Mendel's law with considerable aecuracy, and no exceptions that do not appear to be merely fortuitous were dis- covered" {I.e. p. 15). Apart ft'om the great uncertainty involved in the use of " hairy " as a definite category alternative to " glabrous " (which will be dealt with later) these experiments are of interest, because a parallel series of crossings had already been carried out by Professor de Vries, who nsed the same glabrous varieties as those used, at least in some cases, by our authors. They express theni- selves as " specially indebted to Professor de Vries" for them; and they add in a footnote, " The discovery by de Vries of a wild specimen of L. vespertina var. " glabra, and the artiticial production from it of a smooth red-flowered form are "described in his 'Erfelijke Monstrositeiten,' p. 10 " Now I have not been able to find a work by Professor de Vries, under this title, which contains a page 10 ; but in the Botanisch Juarboek, Jaargang IX. 1897, pp. 62 — 93, there is a paper by him entitled "Erfelijke Monstrositeiten in den ruishandel der botanischen tuinen." Under the heading Lychnis vespertina glahra I find the following State- ment (p. 71): "In August 1888 I coUected seed of Lychnis vespertina in a wood "neai- Hilversum. Among the plants produced in niy experimental garden in " the following year were some completely glabrous exaniples. I isolated these, " and in the course of some generations the glabrous race has been rendered "stable (geheel standvastig) as a result of coiitinued selection (door voortdurende " selectie). It seems never, or only very rarely, to pi'oduce atavistic individuals. " Traces of hairs are still to be found here and there, especially upon the young " plants. This point is worth closer investigation." This race of "glabrous" plants was therefore established by a process of selection, and seven or eight years after it was first observed it still produced traces of hairs. There is no evidence in the account given to show whether the .seed came originally from a glabrous wild plant, as suggested by Mr Bateson and Miss iSaunders, or not; but they may have fuller information from Professor de Vries. In auy case the first glabrous plants observed did not behave as " mutatious," but as we should expect extreme variations from the hairy type to behave ; so that several generations of selection were required in order to fix a stable race with their characters. Although its establishment by continued selection suggests that this glabrous race had not at first the properties of a " recessive " race in Mendel's 48 On MeiideVs Categories seuse, yet it is said to have beliaved as a strictly rocessive foriii when crossod witli hairy races later on. In 1892 it was crossed with normal forms ot Lijchnis diiirna, and the resulting generations were held to obey Mendels laws. The hybrids of the first generation wcrc all hairy ; and the offspring obtained by pairing these werc in part haiiy and in part glabrous. From tiie glabrous hybrids a stable glabrous variety is said to have been raised, and there is no record that these glabrous " recessive " forms ever give rise to " hairy " plauts. The gametes of the glabrous Li/chnis vespertina, as fixed by seleetion, are therefore said to behave in this cross as if they were what writers on Mendel's work call " pure." Mr Bateson rejects the view that the characters of cross-bred individuals, derived in part from such "pure" parents, can be regarded as depending upon the characters of the ancestors from wbich the "pure" parents are descended ; he declares that they depend entirely upon tiie characters of the " pure " individuals used in making the cross. It is therefore to be regretted that he has abstained from discussing an expcriment in which Professor de Vries crossed this "pure" glabrous Lychnis vespertina with Silene noctiflora, and obtained hybrid offspring which were indeed hairy, but their hairs were of the type proper to Lychnis vespertina, and not of the tj'pe of Silene noctißora. Surely we have here a clcar proof that the " dominant " chiiracter, hairiness, may on the application of a suitable Stimulus be manifested by the fertilised germ-cells of what is said to be a purely recessive plant ; so that the theory of pure parental gametes, on which Mr Bateson lays ."^uch stress, is shown to be inade(iuate for this ca.se, and a theory of inheritance, with rever.sion to particular ancestors, is indicated as likely to express the facts of " Mendelian " cases also. As has already been said, Professor de Vries crossed the glabrous L. vespertina here doscribcd and normal hair}' L. diurna, always apparontly using L. diurna ? X L. vespertina ^ ; the first generation of hybrids contained only hairy individuals. The secoud generation is not very fuUy described ; in the Erfelijke Monstrositeiten (p. 72) we are told that about § of the individuals were hairy, and ^ glabrous ; in the Comjdes Rendus, March 1900, we are toid that 28 per ccnt. were glabrous; and this Statement is repeated in subsequent accounts. Although the actual numbers of individuals are not given, so that the probable errors of these results cannot be calculated, it is clear fi-om the adoption of the round numbers i^ and § that the imprcssion produced was not that of \ recessive and £ dominant or dominant-hybrid individuals, as it should have been on Mendel's hypothesis. Several hundred of individuals are said to have been observed. The odds against a deviation so largely exceeding Mendel's rcsult with 500 individuals would be about 17 to 1. The glabrous variety produced in these experiments appears to have furnished some, at lea-st, of the glabrous, rcd-tlowered forms used by Mr Bateson and Miss Saunders, aud in their hands it is said to have given results in good accord with Mcndd's law ; they pass over the deviations from Mendel's law, observed during its earlier history, without notice. W. F. R. Weldon 49 So miicli l'or what is known alxiiit ihu previuus history of the raccs used by Mr Batcson aiul Miss Saundeis. It is sufficient to show that even the use of Mendcr.s categories is not enoiigli to bring the phenomena described into any- thing like e.ract accord with Mendel's laws. Betöre going further, wo will exainine a little more closely the conditious included uader the two categories "glabrous" and "hairy." By glabrous Professor de Vries understands (at least when speaking of L. vespertina) the abseiice of perfect hairs from all parts of the plant, although he calls a variety " glabrous " when especially in the young state it has " traccs of hairs " here and there. Mr Bateson and Miss Saunders leave ihrlr nieaning a little doubtt'til. ()n p. 16 tliey say "in the glabrous varieties no hairs wore observed on any part of the plants at any time"; but since when speaking of cross-bred forins they always rofer to hairiness as a character of leaf-surface, it is not quite clear to nie whether a cross-bred plant with glal)i-ons leaves and a hairy stem was ever observed by them, and if so, whether they would classify it as glabrous or as hairy. The only glabrous plant I have myself foiind wild {L. vespertiiia) had uo hairs on any part of its surface ; and I am not aware that plants with hairy stems or calices and glabrous leaves have been described. For this reasou the repeated statement that hairiness is a character of leaf-surface is difficult to understand. The category " glahrous" whether merely applied to leaf-surface or to the whole plant, certainly iucludes only a small ränge of conditions. It is far otherwise with the category " hai7'i/." In accordance with the differences in habit of the two species, the hairs of L. diurna and of L. vespertiiia differ considerably in their distribution, even when leaf-surface only is considered ; and while the average condition of the hairs in the two "species" is ilifferent, each "species" varies from race to race, and from individual to individual. The hairs are multicellular, and niay vary in length, on the same leaf, from aboiit 0'2 nun. to about 19 mm.; the cell sap may be red, or hlue, or nearly colourless ; some of the hairs on a leaf may be glandulär, the percentage of such hairs varjung greatly. Further, when a race of L. diurna and one of L. vespertina, both being hairy, are crossed, the liybrids are said by Gagnepain to be iutermediate betweeu the parent races in the condition of their hairs {Bull. Soc. But. de France, T. XLiv. p. 44-5). In Order to avoid the effects of differentiation among the leaves, it is necessary to compare corresponding leaves of the different plants examined. To illustrate the Variation in the frequency with whieh hairs occur in Unit area of leaf-surface Table I. has been compiled. It shovvs the number of hairs per square centimetre of epidermis on tlie lower side of a leaf from the node below that which bears the terminal flower. The observations were made (a) upon a race found in a little copse on the Berkshire (right) bank of the Thames, about four miles by road below Oxford, and close to SandforJ Lock; (h) upon a race found on the slopes of Cooper's Hill, Surrey, also on the right bank of the Thames, but more than 35 miles distant Biometrika ii 7 50 On MendeVs Categories from Sandford in a straight line. Tlie second race was collected, and in large part examined by Miss C. B. Sandei-s, whoni I giadly take this opportiinity of thanking for the time and labonr she was kind enough to give. The epidermis was removed frora the Icaf, and mounted in water, with its hair}' side upperinost, the hairs being then counted under a low power of the niicroscope ; in this way error due to the accidental breaking of hairs was avoided, because the stump of a broken hair was easily recognised. TABLE I. Number of Hairs per Square centimetre of Lower Leaf in Lychnis diurna ? . The area chosen in all cases so as to exclude the great veins. Number of Hairs Frequency Number of Frequency Sandford Cooper's Hill Hairs Sandford Cooper's Hill 0— 24 25— 49 50— 74 75— 99 100—124 125—149 150—174 175—199 200-224 225—249 250—274 275—299 300—324 325 349 1 2 5 5 7 17 9 7 9 7 3 7 4 1 10 14 19 17 8 5 3 2 2 3 350—374 375—399 400—424 425- 449 450—474 475—499 500-524 525—549 550—574 575—599 600—624 625—649 650—674 675—699 6 1 6 1 1 1 ? 1 1 — Totais 102 100 The figures given do not coiivey an adequate iinpression of the ränge of varia- bility throughout the species. Eveu in iocalities close to those inhabited by the races described a greater ränge of Variation has been observed ; such as they are, however, the two races exaniiued are enough to show how wide is tl)e range, covered by the category "hairy," in L. diurna alone. In a single race of Z. vesper- tina fron) Sliotover Hill, near Oxford, it is even greater; 112 plauts of this race included one glabrous individual, and one with 1106 hairs per Square centimetre on the Standard leaf The distribution of hairs in tliis race is sfiven in Table II, which is again an inadcqnate pieture of the ränge throughout the species, even in the neighbourhood of Oxford. Odd plants gathered by the wayside have given over 1300 hairs per Square centimetre. W. F. 1{. Wkldon 61 TABLE II. Numher of Hairs per sqiiare centinietre on Lower Lcaf SiirJ'iice in Lychnis i'e.sjiertina $ frinti Sliotover. Number of Hairs Frequency Number of Haira Frequency 0— 24 1 500—524 5 25— 49 525—549 4 50— 74 550—574 75— 99 1 575—599 3 100—124 2 600—624 3 125—149 4 625—649 2 150—174 2 i 650—674 1 175—199 6 i 675—699 2 200—224 5 700—724 2 225 -249 4 725-749 250—274 7 750—774 275—299 5 775—799 1 300—324 5 i 800—824 325—349 3 ; 825—849 350—374 9 1 850—874 3 375-399 7 875—899 400—424 4 j 900—924 1 425—449 9 ! 925—949 450—474 5 ! 950—974 1 475—499 4 975—999 # * * 1100—1124 1 Total 112 Taking a group of 25 hairs as the unit of measurement, and calling the group containing from to 24 hairs the first, we have, for the constants of the three distributioiis the follovving values* : TABLE IIL Constants of the Frequency. Distribiitiini nf Hairiness. Mean Mode 0- /^3 M4 ^1 ß2 "l ("2 L. diurna ? L. Vespertina ? Sandford Cooper's Hill Shotover Hill 10-2255 7-0284 4-8863 132-0653 2658-4536 1-2H1485 4-663582 4-4337 92-7746 1-1689 24-4607 5-4900 2-7725 16-7143 14-4399 7-9583 441-4132 15675-9832 0-766935 3-907930 13-6382 158-3939 2-5372 33-6951 * Notation is that used by Pearson : Phil. Trans., Vol. 186, A, p. 307 et seq. 7— -2 52 On MendeTs Categories The probable error of all these values is of course consiilerable, but they tit the observations as well as could perhaps be expected. Thus the lower limit of hairi- ness in the race of L. diurna froin Sandford is 7-0284 - 4-4337 = •2-5947 units, or allowing for the probable error of the start of the ränge, which is about -03 units, 52 + -16 hairs, the lowest ob.served number of haiis being 47. I feel böiind to call attenti<3n to this feature of the distribiition, which is some sign Ihat in L. diurna the ränge of Variation, in a normal hairy race, does not involve an occasional pro- ductioii of glabious individuals; it is iuteresting in this connection to notice (1) that Mr Bateson and Miss Saunders carefully refrain from speaking of a glabrous variety of L. diurna, and (2) that the red-flowered hybrid, which Professor de Vries calls Lychnis diurna glabra, results froni the cross between glabrous L. vespertina and hairy L. diurna already alluded to. The lower limit to the distribution in L. vespertina is 14-4399 — 13'63S2 = 0-8017 units, or 8 hairs per centimetre, which is as good an approximation to a lower limit at no hairs as we need expect from the limited number of observations. It must be remembcreil that the first unit of the grouping adopted has a centre at 13 hairs, whilc the only individual in the group was glabrous. The Cooper's Hill material secms heterogeneous, and I have not thouglit it worth while to resolve this small number of individuals into componeuts. TABLE IV. Number of Haiis with Glandulär Extremities per Hundred Hairs observed. L. diurna $ L. vespertina $ Gland Hairs per cent. Sandford Cooper's Hill Shotover Hill f ° 89 100* 43 Lcss than 1 3 — 12 4-9 ] 1 — 1-9 2—2-9 3 1 — 2 2 r 3—3-9 1 — 6 . 4—4-9 2 3 5-0— 9-9 1 — 16 10-0— U-9 1 6 150— 19-9 1 5 20-0— 24-9 — 6 25-0—29-9 1 30-0—34-9 350— 39-9 40-0—44-9 — 1 Totais 102 100 103 * So my notes say. If we make tbe utmost allowance for possible brenkage of the tips of ßland- hairg, so that they cannot be recogniseii, it is I think certiiin that no female plant from Cooper's Hill had one per cent. of such hairs even near the base of tbe leaf. W. F. I{. Wki-don 53 The foregoing facts sliowtlu: laiitfc of vaii:itiön in thc nuinber dl' hair.s per unit area cm tlic leaves of a fcw iiidiviiliials. Il is dilHiuilt. io estiinate the percentage freipiciicy of irlaiuliilar hair.s, hecaiise tliey are inore trequent near the base of the leaf tliaii near its apox; I have atteinpted an estiinate of their ))ercentage freijuency in the basal part ot tlie leaf, and the result is given in l'able IV. The few data here brought together are sufficient to sliow tlie way in which the adoption of such a category as "hairy" eonceals the facts of Variation within the races disciissed. In the hght of such facts the statenieiits tiiade by Mr Bateson and Miss Saunders are seen to be utteriy inadäquate, either as a description of their own ex])eriinents, or as a (K-nionstratioo of Mendel's or of any other laws. Whcn hairy and giabrous plants were crossed \ve are told that "aniong the whole numbcr of jjhvnts raiscd, not a singlc interniediate was observed" ( I.e. p. 1 ö), but we are not told what "an intermediate" is. The authors nuist know, or ihey could not certity its absence ; it would have been well if they had thought tit to define an " intermediate " in their Report, for the definition is a vital part of their argument. The prececling tables show that " hairiness " is not an absolute, invariable quality, but that it is manifested in various degrees. In the few individuals exaniined it is possible to pass by a series of small steps from the giabrous condition through individuals with various nunibers of hairs per Square centi- metre of leaf-surface, up to a condition of very great haiiiness. It is perfectly legitimate to regard those individuals with a small number of hairs per unit area as intermediate between those with a larger number and those with a smaller number or with none. Thus a plant with only 80 hairs per unit area of leaf- surface may be called intermediate between a plant with 200 hairs per unit area and a plant with none. Other conditions are conceivable, intermediate between that uf a plant with a given number of hairs per unit area, the hairs being of kuowai length, and that of a giabrous plant. Thus the transition might take place by a reduction in the length of the hairs through various steps to zero, without reduction in their number ; in which case the plant with the shorter hairs would be intermediate between the plant with longer hairs and the giabrous individual ; or again the transition might conceivably be etfected by the appearance of giabrous patches, so that the intermediate individuals assumed a mosaic character. On the whole, however, the density of the hairs per unit area seems the best measure of "hairiness" for our preseut purpose. Whatever we may choose as a measure of hairiness, Mr Bateson and Miss Saunders give no evidence by which we can judge the result of their work. Thus we are told that "haiiiness" is dominant, becansc "of the thousand cross-breds " raised from various unions between hairy and giabrous strains, all, without e.xcep- " tion, were hairy" {I.e. p. 19); but unless we know how hairy they were, we cannot judge what is the valuo of the Statement that hairiness is dominant. Thus a plant 64 On Mendels Categories with 40 Ol- .30 hairs per s(]uare ceiitiinctre is obviously and unquestionably hairy ; it is just as clearly intermcdiate between a plant with 1100 haii-s per centimetre and a glabrous plant. Without some proof that the mean nuniber of hairs per Unit aroa was not less in the offspring of a cross between a glabrous and a hairy form than it was in the hairy parent, the result described cannot be distinguished from that most probable on any theory of "blended" inheritance. It is most unfortunate that the degree of hairiness of each individual was not recorded, beoanso the great variability of the parental races would have given an excellent opportunity of donionstrating the trutli of the statemeiit that pure domiuants "are not merely like, but identical with the pure parents." We might at least have hoped to Icarn whcther a Li/clniis diui-ivi with 40 hairs per centi- metre, and a L. vesperiina with 1100 or 1200, produce plants with ditlercnt degrees of hairiness when crossed with the same glabrous variety, or whether the presence or absence of gland hairs on the leaves of the hairy parent has any effect upon this "charactcr of leaf surface" in the offspring I Unless all the hairy parents used, of either race, were equally hairy, the first cross-bred generation must either have contained individuals of different degrees of hairiness, or the idcntity between pure dominants and their parents disappears. On the other hand, if the cross-breds and their parents differed in this character we have no way of distinguishing "dominance" from the result of "blended" inheritance, until we abandon the Mendelian categories, and adopt a rational way of measuring hairiness. The ab.sence of glabrous cross-breds will not help, until we know the variability of the hairy race, and the ch.iracter of the hairy parent; for the chance of obtaining a glabrous individual aniong 1000 plants of any ordinary race of either species is admittedly sniall, and tiie variability of a few faniilies, in each of which one parent is of fixed character, will on any hypothesis be less than the variability of the race, and the chance that sucii a series of families would contain a glabrous individual, on any theory of inheritance, cannot be estimated without information which is at present not available. The difficulties whicli arise from iniperfect description when we consider the question of dominance in the first cross-bred generation are equally formidable in the case of later generations. The category hairy is so wide that it is impossible to judge how the individuals included in it resemble or differ from their parents. The total resulta of the various crosses recorded are in better agreement with Mendel's Statements than is usuai, but the published data do not afford material for discussing the question how far any particular theory of inheritance can be successfully applied to them. It is deeply to be regretted that so man}' iiitercsting e.xperiments, involving so much time and labour, should be recorded in a form which makes it impossible to understand the results actually obtained, and so gives rise to misconceptions both in the ininds of the recorders and in others. Such justification as there may prove to be for classifying the form of inheritance exhibited by the hairs of Lychnis with W. F. R. Weldon 65 that described by Mciiik'l iiiuy lic just ;is casily secn wheu the variations obsorved both in parents and in otifspiing aii' clcarly described; when this has boeii dmie, and not bcfore, \ve shall be able to consider seriously what the issues raised by Mendel's statements really are. In the inean tiine the accuniulation of records, in which residts are massed together in ili-detined categories of variable and uncertaiu extent, ean h} »3+ - ■Hl + ?!o + «3 + . . . = mean of all capsules in the array of plants due to one parent ... (iii). In this case all we have to do is to correlate the individual capsule with the parental niean capsule ; this correlation will not be significant for heredity, but if we calculate the slope of the corresponding regi'ession line, it will be the slope of the regression line of parental inheritance, as far as it is legitimate to replace (ii) by (iii). Direct test in a few cases showed that with a considerable number of offspring, not oidy (ii) and (iii), but even (i) led to very close results. Accordingly our first raethod will consist in correlating all offspring capsules with the parental mean capsule, and then determining the slope of the regression line — the measure of heredity, free, as we have seen on p. 65, from the effects of selecting individual parents, and also of environmental change of mean. Secondly we may obtaiu a measure of the intensity of inheritance in the following manner. We may compare the avei-age variability of an arra}' of offspring due to a single type of parent plant with the variability of all the offspring populatiou. This method has certain advantages, if we suspect that the enviroumeut of a crop has not necessarily been coutiuuous throughout, aud that 70 Coojyerative Investigations on Plauts sub-environnients inay have altered in an aibitrary nianner the means of the ditferent parent plants. For exainple, with oiily a lew parents somo inay have been more highly t'avoured by light, soil or water than olhei-s. At any rate this method is valiiable for purposes of control, although as it involves the labour of finding individiial offspriiig ineans, it cau oiily be occasionally applied. Thirdly we rnny proceed by the homotypic relationship. This reijuires a brief theoretieal treatment. Let^ be the mean charactcr iu any iiulividiiai parent, P be the mean parent; the mean character in any offspring, the mean offspring and R parental correlation ; then if o-^ o-j, be the standard-deviations of parent and offspring respectively, and iV the total number of cases, we have ^^sip-pno-0) But if we weiglit the offspring as on p. Gl) with the capsule nuinbers of each individual, we shall have R = N'-P)(c-C) •(V), where S' is a summation of every parent jjlant and offspring capsule. But if r be the correlation between parent plant and offspring capsule Hence it follows that ^^S'(p-P)(c-C) iV'o-p'o-c R=^r» (vi). 0"o • Multiplying by irjlap' we have the regression coefiSoicnt equol to ivJiTp the result of cur first method. I. [nheritance in Shlrley Poppy 71 But sinco tho prcsent systeni of weight iiijr niakos and C the saine, wc havo wliere Sc i« tlio moaii S.D. df all the offsprinsf arrays of eapsules = o-c\/(l — p"), if p be the coefficient of hoinotypic correlatioii. Thus wu liave R = rjp (vii). (vii) wouid probably bu a good result to work from if we wanted to find the parental heredity R from r and p in matorial where neither pareiits nor offspiing were selected, and where there was iio niarked chancfc of environmunt between the two generations. Pos.sibly the inflnence of selection of offspring and of their environnient, since tliey affect both /• and p, niay bo le.ss marked in (vii) than in rac/a-p, the rcgres.sion given by the first nicthod, for r and p (.see Table VI.) may tend to rise and fall together, and thus the third method may in some cases give US better results than the first. TABLE V. Parental I nheritance. First Method. Crop Correlation of Parental Mean and Offsprint; Capsule Slope of Regression Line Highgate ... Oxtoi-d Bookham ... Enfield I ... Kiddermiuster ■3230 •1960 •2199 •1864 •1220 •5451 •4064 •3412 •2595 •1589 Mean •2095 ■3422 (7) Results for Parental Inheritance. We have seen that the iirst method ought to eliminate the eftect of a selected parentage, but that it woulil not be uninfluenced by a selection of offspring. There can hardly be a doubt that the Order of Table V. is practically that of the stringency of environment for the five crops, — the most starveling crop being the Highgate oae and the most flourishing the Kiddermiuster. Or, it would seeni that the more luxuriant the crop, the less intense is the strength of heredity. The mean value of the slope of the regression line is not far from the ^ originally given for parental heredity in human stature by Galton, but it is considerably less than the value (about •45) recently obtained for about sixteen cases in man*. We have already pointed out that the restriction of the variability of the general population to be found in an array of offspring due to a single parentage might still be maintained, if different parents had received different treatments. With the assistance of Marie A. Lewenz this point was investigated. She Struck the means of all the eapsules on each of the 907 Enfield I plants. These plants were then grouped into familics and thcir * Unpublisbeil Fauiily Measuremeuts ou upwaids of 1000 famiües in the posaession of K. l'earsou. 72 Cooperatice Investigatlons on Plauts iiKÜvidual Standard deviatioiis as fainilies takeu, as well as the raean and Standard dcviatioi) of the population of 907 plants. The niean of the plant means was 1375 Stigmata, and their Standard deviation 1'370. The average variability (or S.D.) of plants tVom the same parentage was 1-272, and the weighted niean Square deviation — i.e. \fS{na-)!N, where a is the standani deviation of a fraternity of n, and N the total nuniber of plants — was 1287. If /• be the pareutal corre- lation 1 Vi — r', S being the total plant Variation, should be that of an array of offspring. Hence equating this to r287 we find ?• = '3427. Had we equated it to 1'272 we should liave found r = '.3716. In either case this, cur second niethod, gives a value much closer to that ''.i^QO of the third niethod, than the valiie "2595 fouiul for the slope of the Kiifield crop rcgre-ssimi line. Fiiially to illiistrate the second niethod niore completcly a correlation table has actnally bcen formed for the nieans of parent and otfspring plants in the case of the Enfield I crop. The value of the parental nican plant is 12(39 of the offspriiig plant, 13"75, of their respective Standard deviations H99 and 1-370; tlie correlation is ■1561 and the regression coefficient of offspring on parent plant "1784. Nor is this result really to be much wondered at, f)r not only are the Mmeans of the stigmatic bands much infiuenced by the number i 75 influences in tlieir Mowcrs, is well evidenced by the followiug ru.sults provided by W. F. R. Wcldun froin his Oxford crop. TABLE VIII. Natme of Group dealt with Correlation of Tarent Plant aiul Ollspriug Capsulc Regression of OüsprinK on Parent Plant (a) Early cap^5ulo.s (apical Howcrs) of priiicipal plants* •2323 •4003 (6) Capsules on plaiits — not starveliiigs, i.e. with at least three capsules •2430 ■4050 (c) All capsules on principal plants ■2295 ■4295 {d) All capsules on all plants •1960 •4064 The i'egression therefore is fairly constant, althongh the corrolatiou varies. The relationship as expressed by the correlation is greatest, when we avoid differentiation in either the plants themselves or in their fruits, but such differen- tiation does not widely influence the regression, which for this O.xford crop takes a mean value not very divergent froni wliat has been found for the forearm in man ('-12 as mean of four resiilts). (9) Grandparental Inheritance. Our data for grandparental inheritance are very limited and siiffer even niore than the parental material from changes in crop treatment. Thus the original grandparents were grown at Hampdeu on the top of the Chilterns, the parents at Enfield and their offspring at Kidderminster. There is thus considerable change not only in soil but treatment ; unfortunately we had not the means of carrying on the experiments under uniform sivpervision in the same environment. From the Kidderminster crop, we had 19,204 capsules due to 100 parent plants and 24 grandparental plants — in all cases the polleu plants being tmknown. The grandparental plant mean was 12'64 and the Standard deviation 1*1609. The grandchild capsule mean was 13"19 with a Standard deviation of 1"7526. The correlation between grandparental plant mean and grandchild capsule was •0490. The regression of offspring capsule on grandparental plant mean was thus "0739. If we use the third method and the value of the Kidderminster homotyposis it would be '1216. The only result with which these can be compared is the Enfield II grandparental capsule result (see p. 74). Dividing this by the product of the two Enfield homotypic values, we should have ^1897 for the grand- " On the thinning out oue plant, the principal plant was left in a pot, but later other seed germinated and small secoudary poppies were found in a number of pots. 10— -J 7(J Coo2>eratire Investigatioiis &n Plauts parental relationship of plant means. Oiir meagre results may then be sdinmcd up iü thc foUowing table : TABLE IX. Grandparental Inheritance. Crop Method Grandpaieutal Inheritance Pareutul Inheritance Kidderminster Enfield Regression Homotyposis Homotj'posis •0739 •1216 •1897 •1589 •3031 •3660 These results at least iudicate that, however determined, the grandparental relationship is fairly dose to half the parental. It is much to be regi-etted that we have not wider data, especially on a less luxuriaut crop than the Kidderminster one, for the above results arc very variable. (10) Colour Inheritance. W. R. Macdonell made an claborate Classification of the colour of 1604 flowers divided according to their parentagc into 24 groups. He fornied 13 colour classes, which after consideration were classified into tbree groups according to the intensity of red colouring matter in the flower. The first group embraced all the ränge from dark red to red-whites ; the second the pure piuks and the third all the piiik-whites to pure whites. In this Classification attention was not paid to the base of the petal, but to the middle portion and margin. The foUowing are thc frequencies of the original scale : (1) Red, bordered lilac 2 (8) Pink-white, bordered pink 1) (2) Red 275 (0) Pink-white 174 (3) Red, bordered white 70 (10) Pink-white, bordered white 1.55 (4) Red-white 31 (11) White, bordered pink ... (5) Red-white bordered white 43 (12) White faint pink 76 (6) Pink 022 (13) White 50 (7) Pink, bordered white 106 Total 1604 Classes (8) and (11) had representatives in crops from Seed (ß) and (y). Classes (1) to (5) foruied our first general class, (6) our second, and (7) to (13) our third. This grouping enabled us to use the method for dealing with characters not quantitatively nieasurablc, assuming that the distribution of red colouring matter is an approximately normal one. If a be the Standard deviation of the whole population of 1604 offspring flowers, and 2 the mean Standard deviation due to the arrays having a common parent 2 = hn„) " (S(ni?ij) _ )S(n,»!j??!i7»o) _, /Si(ni»!.,) Thereforc r = Rjp-. Here of course ^f(n.,z,) ^ — — — 4 — — 2 6 5 7 :! ■2 29 3 20 — — — — 1 1 S^ Totals 6 '' 104 187 277 i-2'i 662 66.S i;-i i;7i 1 344 204 ■'■' L':i 1 4338 , / nheritance in Shirlet/ roppn 87 D. Enfield 1. Ilomotyposis. N amber of St igmata on ] ^irst Capsule. QJ n 10 /; 12 1& u ir, ICi 77 is 10 2(1 Totals 3 'j r, _. 2 4 3 3 — — — 18 Cd n III 4 18 24 13 9 — — — — — — 68 ^3 11 18 106 151 179 96 31 10 — 1 — — 592 12 2 24 151 368 465 485 180 47 9 — — — 1731 CG Li 4 13 179 465 1026 1191 625 189 49 8 2 — 3751 IJ, 3 9 96 485 1191 1642 1158 462 183 25 2 — 5256 O 15 :? 31 180 625 1158 1140 678 309 57 8 7 4196 16 10 47 189 462 678 590 277 88 20 4 2365 d 17 9 49 183 309 277 138 55 11 7 1038 18 1 8 25 57 88 55 34 9 2 279 CQ 19 2 2 8 20 11 9 2 1 55 *t-( O 20 - — - — — 7 4 7 O 1 — 21 1^ Totais 18 68 592 1731 3751 5256 4196 2365 1038 279 55 21 19370 E. Enfield II. Homotypoais. a o N Limb er f Stigmata on First Caps ule. 7 8 9 10 11 12 13 14 15 10 17 18 Tütais 7 _ 1 — — — 1 S 3 3 1 — — — — — — — 7 9 1 3 6 6 9 2 — — — — — 27 10 3 6 14 31 22 10 1 1 — — 88 11 1 9 31 50 52 29 7 2 1 — — 182 12 2 22 52 72 64 32 10 — — — 254 IS — 10 29 64 74 67 30 3 1 — 278 u — — 1 7 32 67 64 45 4 2 — 222 15 1 2 10 30 45 20 12 3 1 124 1(1 : — — 1 — 3 4 12 14 5 2 41 17 _ — — — — 1 2 3 5 2 2 15 18 - — — — — — — ~ 1 2 2 — 5 Totais 1 7 27 88 182 254 278 222 124 41 15 5 1244 88 Cooperative Investigatlotis on Plcuits F. Kiddenninster. Humotyposis. Nutnber of Stigmata on First Capsule. •: >■ " /" tl /? /.; 1', /.7 /.■ /: IS 7.0 70 Totals 's _ _ 1 4 ■2 1 4 1 _ — 13 cu 7 2 4 16 11 25 34 26 22 15 10 1 1 — 107 u 8 1 4 10 51 91 134 136 117 80 37 7 3 1 — — 672 ■^ 9 16 51 192 486 692 738 558 266 129 31 — — — — 3159 o 10 4 U 91 486 1922 2962 3450 2884 1732 764 223 44 3 3 3 14582 1 rn 11 2 25 134 692 2962 6686 8886 7958 4791 2290 687 176 12 1 — 35302 1 c 12 1 34 136 738 3450 8886 14740 15598 10350 5065 1753 409 52 13 1 01226 ' 13 4 26 117 558 2884 7958 15598 20082 16128 8453 2945 653 141 31 4 75582 C8 U 1 22 80 266 1732 4791 10350 16128 16722 1 10722 4588 1233 276 76 10 00997 H. 15 — 15 37 129 764 2290 5065 8453 10722 9056 4869 1615 396 132 11 43554 '•S 16 — 10 7 31 223 687 1753 2945 4588 4869 3048 1288 357 123 i 19936 (M 17 1 3 44 176 409 653 1233 1615 1288 678 248 77 8 0433 18 1 1 3 12 52 141 276 396 357 248 86 44 9 1026 ^ 19 — 3 1 13 31 76 132 123 77 44 28 5 533 s SO — — — — 3 — 1 4 10 11 7 8 9 5 — 58 Totais 13 1.7 Ü72 :3i:)iJ 14Ö82 3Ö302 Ü122G 7:.o.2 wyuT 13Ö.V1 iyu3tj ti433 1620 533 58 329840 ^ i ■S .2 S "B o o £ " CS .o a J5 G. Highgate. Parent und Offspring. Offspring PLant. Nuinber of Stigmata on Capsule. 8 '' Ä 9 10 7/ 7? 7.^ U ir. 76 17 ToUis ir.'.—urö 2 1 1 1 1 6 10-5— 11-5 1 — 3 4 2 1 — — — — — 11 11-5 -12-5 2 4 4 12 30 37 38 15 6 — — 148 12-5— lä-5 2 1 8 26 42 43 49 34 20 13 2 — 240 is-n—tti-r. — 1 2 4 ") 13 31 21 7 3 1 88 l',:',^i:.- 1 .') •2 1 17 35 32 13 1 113 Totals 3 3 l'.i 11 r,:', st; 1 1-- i:',;i '^S ;>' ' ■■- 2 606 1 H. Oxford. Parent and Off'spriiig. Offspring Plant. Numbcr of Stigmata on Cap.sule. 4 i 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Totals es 10099~10-599 - \ 1 \:r U 16 22 \'.> U 13 -2 1 1 „ 118 s> 10:599— 11-099 — 1 1 5 8 9 17, 21 lOi 10 9 3 — — 106 s =ß 11099— 11-599 — — 5 11 19 35 32 43 48 36 10 3 — 242 ßt o 11-599—12-099 5 13 28 35 53 63 94 81 85 53 31 12 4 3 2 1 563 12-099—12-599 — 10 20 40 59 97 111 122 98 55 25 6 2 1 646 l-s 12-599-13-099 — 1 6 12 23 49 87 120 127 101 72 22 6 1 — 627 £ ^ 1.3-099—13-599 — — 2 4 4 9 10 17 10 20 13 6 3 1 99 ^ 13-.->99— 14-099 — — 1 10 5 15 22' 37 50' 34 12 2 1 — — 195 a s l.>,-099— 14-599 — — 5 12 « 10 10 11 -2-2 14 10 \2 3 — — 132 ^ Total.s 5 26 83 143 195 309 408 409 469 331 189 71 20 C 3 1 2728 I. I nheritaitce in Shirlef/ Po2>2»J 89 I. Bookhiitn. Pareiit and Offspring. i^ z Offspriiig Ph llt. Nuiuber of Stigmata on Cap.siile. 6 7 S 10 n 12 13 u 15 16 17 18 19 20 Totais 9-7o—10;.'ö •2 •2 4 15 IS 19 20 23 13 5 1 — — — 128 10-25— 10-75 — 1 4 11 13 18 18 33 28 19 12 3 2 — — 162 10-75—11-25 — . 1 1 4 4 4 2 4 — — — — 1 — — 21 11-25—11-75 1 3 13 23 27 37 35 25 18 7 — — — 190 11-75—12-25 1 2 6 16 25 40 50 46 45 20 8 2 1 — 267 12-25—12-75 — 1 7 23 30 38 35 29 19 20 9 — — — 212 12-75—13-25 2 11 10 22 4.') 81 83 108 92 49 23 4 — 1 531 13-25—13-75 — — 1 1 4 4 6 7 4 8 1 — . — 37 13-75-11^-25 — 1 3 5 3 11 7 12 11 2 — 58 H-25—H-75 — — 1 4 3 15 28 20 20 22 11 11 3 2 — 140 14-75-15-25 — 1 1 8 10 23 21 25 30 22 12 2 4 — — 159 Tntals 3 11 38 lüG Vr2 238 305 315 302 234 128 50 ; 19 3 ' 1 i:)iiö J. Enfield I. Parent and Offspring. a ^ S ° U 'S ^ Offspri lg Plant. Nullit er ot Stigmata 011 Capsule. S .9 10 11 1.' 1-! 14 15 16 17 18 19 20 Totais 9-75—1Ü-25 _ 7 20 37 71 52 28 16 5 — -" — — 236 10-25—10-75 — — . — — — — — — — — — — — 10-75— 11-25 4 8 12 37 34 19 5 2 2 — 123 11-25-11-75 1 3 3 26 33 75 84 Ol 19 11 1 — 317 11-75-12-25 1 4 14 68 127 186 200 111 58 15 4 — — 788 12-25—12-75 1 3 13 45 04 73 43 23 5 2 — — 272 12-75— 13-35 3 4 34 109 219 308 274 186 74 27 6 1 1245 13-25— 13-75 1 4 26 52 96 120 107 55 25 11 2 — 499 13-75—U-25 — — 1 8 14 44 53 51 17 10 2 1 — 201 14-25—14-75 1 — — 15 22 63 85 72 42 15 5 1 1 322 1 4-75-1 5-25 — — 8 4 20 30 45 44 30 13 6 1 — 201 Totais 3 12 48 222 471 885 1054 810 451 175 60 11 9 4204 Biometrika 11 12 90 Cooperative Inresfi(/afioiis on Plauts a, c c c s I c g, 'S ■-« in <« O = a s K. Kidderminster. Parent and Offspring. Parent Plant. Mean Number of Stigmata on Capsule. ?> "?> o >o o lO o >«3 C) "-'S ?■ >--3 WS 3 95 3 »3 3 3 "o 3 >0 1 O •—1 1 f^ ToUls ^ ■^ 33 **4 '-1 - ■i -^ 6 _ _ 1 _ _ 1 7 — 1 3 1 2 1 1 9 8 4 1 11 5 13 3 5 2 2 3 1 50 9 4 14 14 52 15 66 11 18 10 3 3 2 212 10 22 52 52 289 63 175 62 67 41 27 27 20 897 11 62 151 118 603 151 357 179 202 82 75 54 51 2085 12 75 233 172 995 250 586 384 380 202 148 116 80 3621 13 91 294 209 1133 326 588 470 444 266 261 162 116 4360 U 60 260 184 825 339 469 409 396 297 263 129 122 3753 15 43 135 155 467 22< 235 284 286 228 181 86 74 2402 16 17 55 76 214 Ins 89 133 130 112 99 45 53 1131 17 3 12 33 69 32 20 50 46 23 29 21 37 375 IS — 3 7 18 6 — 6 23 6 9 7 13 98 19 — — 2 2 1 — 4 5 5 7 o 28 20 — — — 1 — — — 3 — — 1 5 Totais 377 1213 1024 4682 1524 2598 1996 2007 1274 1106 654 572 19027 L. Enfield I. Parent and Offspring Means. Offspring Plant. Mean Nnmbcr of Stigmata on Capsules. § V ^ "? 2 i 1 3 i o o S »3 i <>* "5 S3 •-i 1 i 1 i 1 1 >;-3 i 1 7 i — i Tütals 10-005— 10-50Ö 10-505—11005 11-005—11-505 11-505—12005 12-005—12-505 12-505—13-005 13005—13-505 13-505—14-005 14-005-14-505 14-505-15-005 15-005— h'y.'iitr, 1 — 1 1 1 2 1 2 4 1 5 1 2 3 3 9 1 1 1 2 1 5 8 5 7 6 9 6 3 3 3 13 12 6 10 14 3 1 10 12 25 12 31 16 4 4 3 6 10 20 4 24 21 5 7 4 8 21 26 10 40 24 19 8 9 1 12 15 9 26 18 6 5 8 4 9 15 4 39 18 13 3 7 1 8 5 23 7 5 3 4 2 5 9 2 17 4 5 3 3 1 2 10 3 4 1 2 1 3 5 4 1 1 1 3 1 1 1 44 97 154 63 237 148 74 37 40 Totais 1 1 — 1 2 1 a , 16 21 52 1 62 1 117| 101 1 lüJi lOUi 112, JU 50 23 15 6, 1 1X)3 I. Tnheritance in Shirley Poppy 91 M. Od'ford. Stigmata and Lateness of Flowering. Number of Stigmatic Bands on Capsule of First Flower of Plant. / 8 9 10 11 12 13 IJ, 15 TotiiLs Early Flowering... Late Flowering ... 3 1 G 1 11 7 •26 12 30 21 28 41 37 28 U 30 5 18 1 2 161 ICl 'l'otals 1 7 18 :is 51 cy (J.j 44 23 3 322 CM .2P öl ^ (1h S N. O.rford. Pareiit and IJjfspriiujs First Floiuer. Number of Stigmata on First Flower of Offspring Plant. li r tS' :i 10 11 ;.^ 1,! 14 ;.: Total» 15-099-15-599 14-599—15-099 14-099—14-599 13-599-14-099 13-099—13-599 12-599—13-099 12-099—12-599 11-599—12-099 11-099—11-599 10-599—11-099 10-009—10-599 1 2 1 1 1 4 2 2 1 3 3 1 7 1 ■2 4 2 4 10 5 5 6 3 1 5 11 G 10 11 2 4 5 3 8 14 17 9 10 7 9 2 3 16 13 9 1 6 2 4 4 12 9 5 5 6 3 10 5 3 3 1 1 1 2 13 13 31 69 56 50 38 30 27 Totala 3 7 19 39 53 75 57 45 25 4 327 0. Oa-ford Parent and Off'spring Capsules. Oftspriug CapsiL e. Numb er of Stigmat a. i .■> ■ 6 i 8 9 10 11 12 13 u 15 16 17 18 Totais / — — _ 1 4 1 r> 9 4 7 1 _ _ 32 £ f! — 1 / 8 7 19 14 3 G 1 1 1 . . 68 ^ a 9 — 1 5 10 lö 14 17 17 7 9 — 3 99 10 — 2 * 2 16 19 25 14 9 3 1 — !).S 11 3 2 / 9 12 25 13 20 17 9 5 3 — iijri 1> — 3 12 18 21 15 24 16 4 4 3 121 13 — — / 1 19 19 20 18 15 2 4 — 106 a> 14 — • 2 14 16 8 20 13 12 7 4 3 1 — 102 cSl 15 — 1 3 5 4 8 15 17 21 16 3 1 1 96 16 — 1 10 11 9 17 11 14 7 7 2 89 iz, 17 — 1 1 4 14 16 9 33 21 15 2 — 117 IS — — — 9 9 8 18 18 10 6 4 2 1 1 87 TotaLs 3 14 73 1 94 139 181 184 191 128 83 29 14 3 3 1 1140 12—2 92 Cooperative Inrestigafions on Plauts P. Crockham. Parent and Off spring Capsules. VI fco p< -•J rt 7; Ü •^ c: 9 CS -= 'A S 9 10 11 13 13 H 15 16 17 18 TütuLs Ofifspring Capsule. Number of Stigmata. /..■ /! /■: !■: II 1 . IS I 19 21 7 10 8 6 2 4 4 6 1 5 8 19 2(; 44 70 .">f; 18 5 :{ 20 34 42 50 55 25 4 1 15 26 öl 72 52 26' 13 9 30 40 55 72 62 72 49 26 7 22 52 52 52 34 16 9 2 3 7 11 14 15 5 2 9 22 53 41 51 40 29 24 6 6 14 29 26 14 12 5 14 36 50 84 72 35 25 11 — 9 23 33 32 36 22 6 7 26 32 36 60 52 Ml' in 16 16 23 31 35 42 12 :;:i 1 1 14 6 15 3 — 23 IG m , 145 1 260 , 446 OHl , oü; 409 2Ö4 145 I 58 18 8 — , 1 Totala 250 243 280 438 260 62 2S8 119 342 170 261 278 2991 Q. Enfield I. Parent and Offspring Capsules. Offspring Capsule. Nuni 3er f Stigmata. 7 8 9 10 11 12 IS U 15 16 17 18 19 20 21 Totala 7 — ^ - 1 -^ 8\ 12 23 3.") 31 30 18 7 2 1 170 J 8 — 1 — • 1 1 10 19 45 49 37 19 8 1 — 190 9 — — — 1 8 23 29 40 29 22 9 6 2 2 2 173 ■3 s 10 — — 1 7 2' * ■J'.i :',;• 33 17 7 3 1 1 163 S--^° 11 — 2 1 1; ]i; 2:1 l!l 33 18 4 3 — 198 ö^t 12 — 1 2 , 9 13 31 ■).". 21t 38 14 11 3 1 — 197 - IS — 2 2 9 18 21 32 l'l 28 23 8 1 1 2 — 187 £ s 9 H — — — 1 5 14 i« 36 35 32 25 7 6 — — 179 15 1 3 1 13 6 18 26 31 44 28 15 12 3 2 203 16 — — 1 — 9 12 15 27 17 17 13 6 3 — 120 ;z; 17 — — 2 3 3 10 23 22 24 28 14 10 3 142 18 _ _ — ■ — - 1 7 19 39 36 21 14 ö 2 — 144 Totais 1 , . 11 :,[ . uy.i 1 2IU 1 329 1 1 1 1 410 j 3-S8 284 1 153 1 73 , ''\ 10 2 2066 I. /nheritance in Shirfei/ Po}>py 93 R. Enfield 11. Purent und OffspriiKj Capsules. Offspring Plant. Nuniber of Stigmata on Capsule. g 60 3 3 25 i- ü 1 i 19 m Totais' 1 luvißj — tiröoö _ _ 6 16 43 107 1 149 170 157 113 64, 19 3 847 -i i 1U-50Ö— 11-00 ') . — — — — — — — — — — — — — «^ 11005—1 1-505 — 2 7 56 173 354 572 630 432 247 103 17 5 1 — 2599 1 11-505—1^-005 — 1 4 30 130 303 516 527 423 291 172 65 16 3 1 2482 1-3-005—12-505 — 2 1 11 58 187; 286 374 283 173 64 27 3 1 — 1470 12-505—lä-005 1 2 13 45 298 605 1019 1416 1348 880 415 144 44 9 1 6240 So 13005—13-505 — 2 12 54 136 330 533 548 452 308 109 3S 14 4 2 2542 ■? '^ 13-505-14-005 — . — — — — — — — — — p a 14-005—14-505 — — 8 10 36 132 265 336 356 214 123 . i 1 s 5 1 1535 ö-a- 14-505— 15-005 — ' — — ■2 31 66 123 145 97 62 25 5 — _ — 556 M 15-005—15-505 — — — 4 33 89 210 222 196 104 48 16 6 5 — 933 ToUls 1 9 1 51 -2-2^ 938 :.'173 3(i73i 436-i|3744 2:i92 1123 1372 99 28 5 19204 U. Highgate. Brother Plauts. First Brother Plant. Number of Stigmata on Capsule. CS p a ■^s y. ' 7 .s- 9 10 11 1? IJ '4 15 Ifi 17 TuUils 61 1 X 6 8 9 11 7 4 5 9 1 i 1 2 7 5 10 11 6 4 5 9 1 — 61 .s' 6 7 18 45 52 55 52 58 44 35 6 1 379 .'/ 8 5 45 186 404 425 562 490 260 126 13 5 2529 Kl 9 10 52 404 120 655 990 1353 738 287 48 6 4672 n 11 11 55 425 055 874 1593 1586 695 305 29 10 6249 1 : 7 6 52 562 990 1593 21.58 2.331 1227 471 49 28 9474 /.; 4 4 58 490 1353 i.-ise 2331 3230 2116 809 106 63 12150 / ; 5 5 44 260 738 695 1227 2116 1478 640 75 51 7334 / '■ 9 9 35 126 287 305 471 809 640 284 31 20 .3026 i'i 1 1 6 13 48 29 49 106 75 31 6 3 368 1 : — — 1 5 6 10 28 63 51 20 3 187 T..t.ils 1 i;i (;] ■■',''.> ■2'>-2\> 4.72 6249 9171 1 2 1 .M 1 7:i3l 3021 i 3t;s 1N7 KUDO I. 1 nherllaace iit, Shirley Poppij 95 V. Oxford. Brother l'lant. First Brother Plant Number of Sti ^mata on Capsul 3^ .'/ ;- t_i 7 ,S' m // 1.' /.,■ 14 ;.•: 16 17 /.s' /■/ 'l'otals 4 () :39 50 63 68 86 107 95 83 52 3(i 15 4 3 5 1 703 /^ ■2!) 88 210 278 330 422 520 513 383 239 162 55 13 10 15 3 3270 i Ö 50 210 541 774 1001 1364 1631 1567 140:2 825 536 232 62 30 31 9 10265 / 63 278 774 1051 1411 2124 2560 2784 2547 1649 1023 459 124 34 33 8 1 6922 ,s^ fiS 330 1001 1411 1900 2949 3730 3942 3622 2314 1347 541 168 55 47 16 234 U äl = n 8ß 422 1364 2124 2949 4362 5680 6345 5914 3894 2236 891 261 62 46 11 36650 Sl,=e III 107 520 1631 2560 3730 5680 7604 8575 8515 5694 3367 1337 412 106 70 27 49935 -3 II !),i 513 1567 2784 3942 6345 8575 9789 9693 6734 40()9 1641 473 96 49 16 56381 ^4 l:> s:j 383 1402 2547 3622 5914 8515 9693 9916 7108 4065 1541 470 149 67 30 55505 -^■5^ IS 52 23!» 825 1649 2314 3894 5694 6734 7108 4906 2953 1105 345 105 33 14 37970 ■a^J 'i .■iß 162 536 1023 1347 2236 3367 4069 4065 2953 1844 748 225 50 14 5 22680 S° 1.-, 15 55 232 459 541 891 1337 1641 1541 1105 748 306 107 13 6 2 8999 -^s-, l<: 4 13 62 124 168 261 412 473 470 345 225 107 21 4 — — 2689 J3 s 3 77 3 10 30 34 55 62 106 96 149 105 50 13 4 — — — 717 i.^ 5 15 31 33 47 46 70 49 67 33 14 6 — — — — 416 12; VI 1 3 9 8 16 14 27 16 30 14 :, 2 — — — 145 Totais 703 3270 10265 16922 23441 36650 49935 56381 55505 37970 22680 8999 2689 717 416 145 326688 W. Enfield I. Brother Phints. First Brother Plant. Ntimbcr of ,Sti'j;inata on ('apsnle. O O Ü , 'A 8 9 10 11 12 13 U 13 16 17 18 19 20 Totais 8 6 5 44 59 121 132 92 43 15 7 1 1 526 9 6 8 27 160 285 506 577 374 201 56 26 1 — 2227 Kl 5 27 144 430 971 1839 2036 1410, 772 288 116 14 1 8053 11 44 160 430 1884 4014 7915 8671 61481 3276 1276 445 62 12 34337 12 59 285 971 4014 8636 17042 19309 13862 1 7958 2941 1057 158 12 76304 IS 121 506 1839 7915 17042 32776 38389 28060 15712 6102 2061 301 42 150866 H 132 577 2036 8671 19309 38389 45082 34343 19427 7550 2680 386 63 178645 lö 92 374 1410 6148 13862 28060 34343 26808 15581 6256 2172 393 60 135559 16 43 201 772 3276 7958 15712 19427 15581 9108 3597 1363 259 •29 77326 17 15 56 288 1276 2941 6102 7550 6256 3597 1398 532 104 16 30131 IS 7 26 116 445 1057 2061 2680 2172 1363 532 198 42 5 10704 19 1 1 14 62 158 301 386 393 259 104 42 10 1 1732 20 1 — 1 12 12 42 63 60 2,1 16 o 1 - — 242 Totais 526 2227 8053 34337 76304 150866 178645 135559 77326 30131 10704 1732 242 706652 !>6 Cooperatire Inrcstüjations oii Plauts X. Kidderminster. Brother Plauts. Number ot' Stigmata on Capsii le of First B rother Plaut '-• ' 7 S .9 70 7/ 7' 7.? 1 /.; 75 ir, 1 77 7.? 10 20 ToUls (i _ 1 1 5 8 10 34 29 24 18 5 1 136 3 7 __ 4 13 98 231 343 468 364 216 73 24 •j — — 1S39 a: . S 1 4 58 228 632 1300 2010 2192 1782 1037 429 112 31 2 — 9S18 n 1 13 228 822 2709 5339 8569 9330 7333 4275 16871 553 108 17 3 41)987 U) 5 98 632 2709 10668 23028 37310 43163 33885 19733 8836 2735 784 153 2(; 183765 = i. II 8 231 1300 5339 23028 50446 85615 98062 80399 48752 21743 6844 1742 378 61 42394« ^^ jj 10 343 2010 8569 37310 85615 146104 168554 140664 85882 38603 12099 2772 734 1(X) 729369 ^1 IS 34 468 2192 9330 43163 98062 168554 198452 17U465 106716 49780 16708 4250 1063 179 m;i)416 ?r^ U 29 364 1782 7333 33885 80399 140604 170465 153152 98284 46342 15892 4135 1117 210 754053 ^■2 i.5 24 216 1037 4275 19733 48752 85882 106716 9S284 645 U 31968 11064 2944 881 188 470I7'< 16 18 73 429 1687 8836 21743 38603 49780 46342 31968 15848 5820 1620 497 122 223386 C ü 17 5 24 112 553 2735 6844 12099 16708 15892 11064 5820 2320 695 20J 66 75137 t- ^ IS 1 5 31 108 784 1742 2772 4250 4135 2944 1620 695 200 67 20 19374 -3 19 — — 2 17 153 378 734 1063 1117 881 497 200 67 26 1 5142 20 - — — 3 26 61 100 179 210 188 122 66 20 7 2 984 TuUls 130 1839 9818 40987 183765 423948 1 729369 869416 1754053,470478 i 1 223386 75137 19374 5142 984 3S 13832 Y. Oxford. Bruther Plauts. First Brother Plant. Nodal Height in cms. CS ■3 g a m ^ ^ ^ ©j ©» !3 ©j ©j s ^ 1 ^ ^ ^ ^ g 'l'otlls <■-. ',: ^j ^ i ^ ^ ^1 ^i ^ Ä § •^ i ^1 ^ ^ 3 o tj t1 3 - r,-25—l(>-2ö 3 5 8 6 1 1 24 1 10-25— 15-25 170 207 242 111 60 16 8 — — — — 2 816 15-25- 20-25 3 207 584 762 439 213 86 23 6 — — 5 2328 30-25-25-25 5 242 762 888 626 360 186 77 28 5 — 2 11 3192 25-25-30-25 8 111 439 626 540 436 231 124 66 28 — 2 5 2616 30-25-35-25 6 60 213 360 436 468 296 218 98 58 — 8 11 2232 35-25—40-25 1 16 86 186 231 296 166 135 59 32 — 7 9 1224 40-25— 45-2'> — 8 23 77 124 218 135 128 46 31 — 1 1 792 45-25—50-25 1 — 6 28 66 98 59 46 18 8 — 3 3 336 50-25-55-25 — — 5 28 58 32 31 8 6 — — — 168 55-25—00-25 — — — — — — — — — — — — 60-25— 0.5-25 — — 2 2 8 7 1 3 — — — 1 24 65-25—70-25 — 2 5 11 5 11 9 1 3 — — 1 — 48 1 Tütais 21 810 2328 3192 2616 2232 1224 792 336 168 " 24 48 H'-OO I. l7iheritance in Shirley Poppy Z. Oxford. Brother Plauts. First Brother Planf. (':i|isuli' Height in cms. 97 O n t^ I'^' >o !P J? !P -'^ ^"t 1.-5 «3 "o >ci »ii »0 f^ s< ®^ ^ IS< j» sj <^> ö> «5 1^ 00 53 S ■5^ CO ^ CO ^ 00 ^ "< ^ CO ö^ »^^ e.) 30 ^ -^ 10 '•■^ 'O ■0 ii <~ Co Co i8-;^5— ;,«-;.'.T 3 2 5 4 ►7 1 1 1 1 ^~~~ 24 23-25— ^S-JJl — — 8 15 20 24 14 3 7 1 1 2 1 96 28-35— JJ-2.-, — 8 24 85 88 107 72 29 28 12 1 9. 456 33-25— 3S-d5 — 15 85 244 255 270 223 90 64 40 4 4 9, 1296 38-25- J^-i?5 :5 20 88 255 354 424 316 145 144 93 21 14 7 3 8 1 1896 43-25— .'iS-J.: 24 107 270 424 580 376 219 206 119 35 34 12 10 1 4 1 2424 4S-25— Jo-:.'.; .) 14 72 223 316 376 314 205 228 150 36 19 26 19 2 8 3 2016 53-35- 58-25 4 3 29 90 145 219 205 226 254 210 75 54 45 35 2 6 6 1608 58-25— 63-25 / 7 28 64 144 206 228 254 240 200 82 48 53 28 5 9 5 1608 63-25- 68-25 1 1 12 40 93 119 150 210 200 172 67 48 36 40 7 2 2 1200 68-25— 73-25 1 1 1 4 21 35 36 75 82 67 26 27 14 9 3 3 3 408 73-25— 78-25 — 2 — 4 14 34 19 54 48 48 27 14 8 10 9. 2 2 288 78-25- 83-25 1 — — — ' 12 26 45 53 36 14 8 8 6 216 83-25— 88-25 — — — — 3 10 19 35 28 40 9 10 6 6 2 168 88-25- 93-25 — — — — — 1 2 2 5 7 3 2 9, 24 93-25— 98-25 — 1 ■2 2 8 4 8 6 9 2 3 9. 1 48 98-25—103-25 — — — — 1 1 3 6 5 2 3 2 — — — 1 24 Totais 24 96 456 i29(; 1896 2424 20 IG 1608 1608 120U 408 288 216 168 24 48 24 13800 "S ^ 3 S. E w (Jl a. Oxford. Brother Plants. First Flower. First Brother Plant. Stigmata of First Flower. (7 / Ä .9 10 11 12 13 U 15 Totals 6' 2 9 15 14 17 11 10 1 2 81 i 9 12 30 30 26 14 13 3 4 141 8 15 30 44 62 58 50 52 22 9 2 344 9 14 30 62 72 101 106 103 58 29 5 580 10 17 26 58 101 120 156 159 75 50 3 765 11 11 14 50 106 156 166 207 129 67 10 916 12 10 13 52 103 159 207 174 146 70 9 943 13 1 3 22 58 75 129 146 88 52 11 585 H 2 4 9 29 50 67 70 52 36 1 320 15 — — 2 5 3 10 9 11 1 — 41 Totals 81 141 344 580 765 916 943 585 320 41 4716 ß. Oxford. Brother Plants. First Flower. First Brother Plant. Number of Petals ou First Flower. ii Four Petals Four Petals and Five anil more Pctaluiil Stamens Petals Totals TS a Ol Four Petals Four Petals and Petaloiil Stamoiis Five and more Petals .. 3086 118 549 US 2 29 549 29 212 3753 149 Totals 3753 149 790 4692 Bionietrika 11 13 98 5 s 3 CQ fe es '3 s ■c a o I c J M (NX.re'jiOOOsSwTfOI-'Ni.-^MXtNOTCJtDOOSXOKCSl-XIWO— '- _„ ^-.(N.-HCTTf(!q(N(NlMa0.-i— li-ilN'^l"-' CT X 1 1 1 1 l'-'l 1 |-c|t~TOt~.-'TOOi»i.'5ttl^t~T)iOi-ii-'iO-5 (M ^ t-i 1 CT 6: II II II 1 r-. (N (N — M — CT 1-1 — — (N — — F- 1 1 1 CT 05 CT ^ -. 1 1 1 |i-i(N(S(N'* ii-.in-^cOTj.coMot-.-1-ot-cn-^x — «DMioco |(N0 ■* in ^ ^ eo CD CO — ^ 05 T)< t- — X 1> -# ■* — CO 1 1 1 1 1 1 ——CT — CT— — — — 1 1 1 t~ ^ CO 1 1 1 1 ■* lO CD CD X 1 CO CT in — 05 X m o 05 m CO t- CO CO ■* CT o 05 |— — | ■* | | | CT i CT 1 1 1 |eoeo-*-*CD — CTXo — CT05T»iCTr-CT-*oiacot~-<)iTin — — CTinCMCT — — — |CTeo| 1 1 |-^| 1 1 eo § i-.| 1 1 1 |— CTCTCOj- ■^eOlCO^F^jCTCT — — j— 1 1 jCTj— 1 |^| | j s Ol M 1 M 1 1 1 1 1 M 1 ! 1 1 1 1 1 1 ! II 1 1 M 1 M 1 1 1 1 1 1 1 o CO 1 1 1 1 1 1 1 1 1 1 1 1 1 i 1 1 1 1 1 1 1 1 1 1 1 1 M 1 1 1 1 1 1 M 1 o K •^ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 , 1- 1 -CO 1 CT- .,..-.,- - - 1 1 1 2 IS M 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 II 1 1 II 1 1 1 1 o 1 1 1 1 1— — CTCTct|— CO- |ctcO— !— CT— 1 1 1 1 1 1— 1 1 1 1^1 1 1 s •i^ ^ t~ V- ^ ^ "-^ ^( ^ ^^'-"^ '^ f'- V" — ^ — ^ «"^j '^ ^—i- ** i^ V ^ ^ — , ^ i — ' *•—!.* "c i^ ^ r" ^ "^ — •-. >i »c ~i Sl «1 OJ 9j ©j ©5 ©( ej Sj ©; 00 »3 »3 '»5 «0 05 »5 »5 "»S '»3 •«^ -^ ■c «.^ 1 •BftHanjliiUTi ni j9avo[J [BdiDiiijj no |ijio,| jo niSiioq l"«!,] •laino.ig puoodgi 1. Inheritance in ShirJey Poppij 99 S. O.ifurd. ßruther Plauts. First Fimuer. Finst Biothcr Plant. Breadth of Margin of First Flower. ^ .S . 'S * s S So Broud Broad- Slight Slight Slight- None Noue Totais Broad Broad to Slight Slight Slight to None... Noue 122 3 3 — 108 4 21 — 411 6 108 4 86 15 381 21 15 2 46 411 6 381 46 2324 665 13 594 84 3168 TotaL-s ... 665 13 594 84 3168 4524 0) PL, ^ o 5? & m 03 JS e. Oxford. Brotlier Plauts. First Flower. First Brother Plant. Extent of Basal Patch of First Flower. Nene None- Slight Slight Slight- T-, c -, Definite ^'='^°'*'^ Definite- Large Large Totais None None to Slight ... Slight Slight to Detinitc Detiuite Deflnite to Laigc Large 24 21 44 1 62 128 21 12 22 21 46 44 22 86 3 159 6 224 1 3 3 19 62 — 21 , — 159 , 6 3 — 858 19 19 — 582 8 128 46 224 19 582 8 1000 280 122 544 26 1704 33 2007 Tofcils 280 122 544 26 1704 33 2007 4716 c . o 5 S i^ a ^ S '^^ U i*-i m o -g O OD f Oxford. Brother Plants. First Flower. First Brother Plant. Intensity of Wrinkling on First Flower. Frilled Wrinkled Slightly- Wriukled Slightly-Wriukled Not- Not-Wrinkled Wrinkled Totais Frilled Wrinkled Slightlv-\Vrinkled Slightly- Wrinkled to Nnt-Wriiiklod Not- Wrinkled 10 1 89 89 566 75 : 569 1 17 22 1 145 75 569 718 26 275 1 17 26 2 12 22 145 275 12 1.54 197 1386 1663 58 608 Totais 197 1386 1663 58 608 3912 n—1 100 Cooperatim Investigatiotu on Plants . 1^ g EH i: 'S ü S o S (-1 ~ 1]. Odford. Brother Plauts. First Flotrei: First I^iother Plaut. Colour of Middle Third of First Flower. Eed Bed- Pink Pink Pink- White White Pink-White White Totais Rcfl Red-Piuk Pink Pink- White White to Pink- White AVhite 1032 55 387 154 40 11 55 387 ' 154 4 49 6 49 1078 342 6 342 l'^2 1 2(1 h( — 4(j Kn 4(1 1 26 10 2 6 1 1 46 18 6 o i(;79 115 1928 712 85 83 Totais 1(379 115 ( 1928 1 712 1 85 , ft3 4602 9. Enfield II. Plauts froni seed of same Capsule. Stigmata on Capsules of First Brother Plant. 6 7 S 10 /; 13 13 u 15 16 17 IS Totais t ö ■2 1 2 4 9 s / — — 1 — 4 8 5 7 3 1 29 CO S — — 4 10 18 37 21 23 14 4 — — 131 Cm o ^ 9 — 1 10 24 38 92 68 70 50 26 6 — — 385 10 2 — 18 38 124 270 158 137 96 37 2 1 — 883 "^s 11 1 4 37 92 270 472 358 321 259 119 26 13 5 1977 p. .3c'. Cross 8. ? pure-brcd albiuo. (J Japanese waltzer. Young. 5 c. Gross 9. ? cross-bred albino. (J Japanese waltzer. Young. da. Cross 12. ? piire-bred albino. 6 Japanese waltzer. Young. 4 e. 2d. Cross 13. 9 pure-bred albino. 6 Japanese waltzer. Yoanij. 2 6. 2 f. Cross 16. ? cross-bred albino. (J Japanese waltzer. Young. 6b. Gross 20. 9 pure-bred albino. 6 Japanese waltzer. Young. 4 c«. 16. Ic. It will be Seen by reference to the above figures that there are 18«, 13 6, 15 c; and 2(1, that is to say, 31 cases at Icast out of 48 in which albinism is not recessive ; and even in c, which is a gray mouse superficially not unlike the house mouse, we by no raeans find a coinplete disap- pearance of whiteness ; for as it has already been said, their bellies are nearly white ; which is not true of the house mouse : and the bellies of d are like those of c. 104 Miscellanea The Influeme of Ancestrij. The above tables show that hybrids with the coloiu- of the house mouse all over (except the lielly) appear only in the litters of pure-bred albinos, i.e. 3c in Gross 7, and 5c in Gross 8, 4 c in Gross 12, 2 c in Gross 13, and 1 c in Gross 20, whereas they do not apj^ear in the litters of cross-bred mothers. Of coui-se so small a number of trials does not prove a definite rule : bat the resiilt observed may be connected with the fact tliat von GuaiUi who used albinos in-bred for 29 generations always got hybrids identical with the house niotise. And it looks as if it could be Said that the niore in-bred an albino is the less power it has of transinitting its whiteness. But many niore crosses must be made before any definite statenient can be niade on this point. / ^ Fio. 1. m^ -^, riä^:^ Fio. 3. ,•'*?•' Fig. 4. Miscdlanea 105 II. Interpolation by Finite Difierences. {Tino Independent Variables.) By W. I'A[JN elderton. Let it lio reciuired tu lind «,,,. in terms of «qo, m^^j, ... «j.j ... m_ju ..., a^._^..., wliere ^ and r luc both <1. «p„ = (l+Al)""0 :r = (l+A,)''(l+A,)'-M„,„ = 1 1 + (7'Ai + J-Aj) + — (^)(=) A,2 + 2pr A, Ao + )-(-lA/) + ^,(/.P)Aj:' + 3/.l%A/^A, + 3/>rC^)A,A/ + rP)A,/)...JM„,„ (1), where /)(")=/? Q>- 1) ... (jo + ?i - 1) and A, and A.^ rei)rcsent opcration.s with resiiect to ^; and r respectivcly. If we \i.se the expre.ssion {1 +(/iAi + rA2)j ?/„,„ for u,,.,. we reqnire only 3 value.s of tlie fiinction, while if we take in the next term in round brackets we require 6 values, but the objeotion to the t'ormuUe .seenis to he that the Viihies of the function which we use are not necessarily the nearest vahies to Up.,. For practical purposes it would be better to have the expression in terms of the function rather than in terms of the dift'erences, as the calculation of difterences running in three directions is troublcsome and tlie work i.s likely to contain sHps*. The following scheme shews the form of the problem and gives an idea of which are the best functions to use for interpolation ; •»l:-l «1:0 Mi:i «1,2 «2.-1 «2:0 «2:1 "2:2- NOW «,,-0 = «0:0+.P(«1.0-«0:0). «P:l = "o:l+;'(«l:l-«0:l) and interpolating between these vakies we get (when s=\ - r and q=\ - p) «p-r = -6)(jj-o)... (p-a){p-c)... """-{a-b) (a-c) ... "^ {b -a) [b-c) ... ''^- [* The biometrician has ri^peatedly to use tablos of double entrj' : e.g. in tlie eases of skew Variation ■when using the G-integral (Brit. Axsoc. Repoil, 1899), in dealing with goodness of fit (Biimutiiku, Vol. I. p. 155), iu finding the iufluence of natural selection on conelation {Phil. Trans. A, Vol. 200, p. 64), etc., etc. Hence the iniijortance of a good iiiethod of interpolation. Ei>.] t See note by T. G. Ackland, Juunuil Iitstiliilf of Actiiurii-.i, Vol. 32, p. 2815. Biometrika ii 14 106 Miscellanea and put a = 0, 6 = 1, c= - 1, rf = 2, e= -2, etc., we have „ (p-l)(j> + l)(y-2)(j> + 2)... ^ p(jo+l)(y-2)(;. + 2).. y(f)- 1)0>-2)( ;; + 2).. "'= (-l)(l)(-2)(2)... "»+(l)(2)(-l)(3)(-2)..."' + (-l)(-2)(l)(-3)(2)... "^ " =^^L«n + ,. JfL + ,., 'fzL + ... + ^^ + ...l (3), where it is assumed that 2,11 terms are used, so that ;,,,„=(^ + 7i-l)...(i)-w) and o, = (- 1)»-' j^^^^l^^y^^. These two functions, p.>„ and e„ can be calculatod ejisily as in the csise of a two-variables inteiiio- lation n will not he greater than 2 or for a one-variable thau 4. Taking ji = 4, < = 2 as an example _ 1 _ 1 ''~[2[5 ~240" For the two- variables we shall use (3) for obtaining an interpolated value in tho foUowing form and .(4). By this method we see the coiinection betwoeii Professor Everett's Central Diflerence* and Lagrange's formula;, for if we write the differences in the foruier in terms of the function and take p^n outside a bracket we see that the two are alike. In the actual work in the case of a two-variables interpolation p.,„ x r.j„ could easily bo obtained and tlien we could find the coefiBcients -i-^2nX''2n- Such reduced coefticient for i(, ., woukl be e,c, {p-t){r-s) Let US find as an example of the formulne an interpolated value for ages 51 and 28 from the foUowing values at 3°/„ interest from the //" Joint Life Annuity Table t. Age of Younger Life Age of Eldei- Life 45 50 55 9-6503 9-5902 9-4855 9-3536 20 S5 SO S5 14-1930 14-0131) 13-7313 13-3625 12-8092 12-G787 12-4090 12-1954 11-2791 11-1886 U-0378 10-8436 ' n au | (g + ^)''(g-^) a ,( g + 2)-(7-2 )„ , , (P-H)P(P-1) . (p + 2)...(p-2) where a^, a^... are cven central differences of ii,) and fco, b,... tho.se of i/,. — See Journal Institute of Acluariei, Vol. 35, p. 452. t //■" is the nnnie given to the raortality table constructed by the Institute of Actuaries (lHfi9) from the experience of licnltLy male lives assurcd by Knglish oflices. Misccflaiiea 107 Wo olitiiin tlie fuHuwinj,' valuos By (1) Ist Ditroreiico.s (:5 valuea) 12-2549 (1) 2nd „ (6 valucs) 12-2773 (2) 4 values 12-2G19 (4) Iß valuo» 12-2815 The valuo giveii in tlie Tables i» 12-2811. In applving forniula' (4) if we usc 2» toi-nis in oacli equation \ve, require 4«- terms altogether — thuM 4, IG, ;5G... are the nunibers of turni.s that will uc-cur. üf cnur.se if wo usc an odd number of terms, aay 2« + l, the fcirnuila roads '2.. + 1 "-"+••• +-^ '+... and u„,r = r,„^A\'^+...+J-'ctc.\ «;-.r = P2.. + l| - +---+7_; + ---J- •^"" "7.:r = '-2,.+ lp- + ----*-— ^CtCj where p„a + i = (p + n) (p + n-l) ... (p+l)p(p-\) ...{p- n). The formula; can also be used when m terms are used in one direction and // in the otlior. As an example of the füll working take the table of joiut life annuities given and find the annuity on two lives aged 51-27355 and 28-63984. Here p = -25471, )-=-72797. log(ji) + l) -0985433 log(2-r) -1044973 log (2-/)) -2418676 log (»•+!) -2375362 log {p) T-4060460 log(l-?-) 1-4346168 log (1-ju) 1-8723253 log(r) T -862 1135 \og p^r^ 1-2575460 'o = 2 ' '-'l = ^ 5' "^2 = iS' '' - 1 = ~ ff ' log 4 = 1-6989700 and log J = 1-2218487, and we get the following e.'cpressimis corresponding to {...} in (4), where [-2...] represents antilog -2.... "p" terms -[T-1233054] ';_i + [ -2929240] »„ + [1-8266447] »,-[2-9799811] Hj, ";•" terms -[2-9843125] «_, +[1-836850.5] «„ + [ -2643532] »,- [T1173514] «j, ( + ) 1-1520924 (-) 1-1075220 (-) 10512745 ( + ) -9845408 2-1076179 T-27723G5 2-8109572 r5-9642936 (-) 1-1465311 ( + ) 1-1030747 ( + ) 1-0487757 (-) -9818277 2-9601619 -1297805 T-6635012 2-8168376 (-) 1-1377117 ( + ) 1-0958316 ( + ) 1-0428827 (-) -9770602 1-3876586 -5572772 -0909979 T-2443343 ( + ) 1-1258878 (-) 1-0861960 (-) 1-0351735 ( + ) -9709788 2-2406568 T-4102754 2-9439961 2-0973325 The ui)per line in cach ca.se gives the logarithni of the Joint life annuity, and the lower line the sum of the logarithuis of the coeffieients shewn in the two expressions marked "/)" terms 14—2 108 Miscellaiiea and " )■ " terms. The two lincs are added at sight and tlio anti-logarithm foiuul. The sign of each expression, given in hrackets, is a great help to rapid work. •18185 1-27847 •23257 3-35251 17^09435 2-42526 44-98925 3-13669 5-15557 •72984 13-61071 -95317 •08889 •62902 •11703 1-66492 81 ^4 7022 14-16988 14^16988 log 67^30034 = = 1-8280173 I^2575460 10855633 Using the aunuities for 51, 28 ; 51, 29 ; 52, 28 and 52, 29 with formula (2) I get 121775. III. Variation in the Moscatel {Adooca Mosel lateüina, L.). By hexhv whiteiieau. The mode of arrangoment of flowers in globose hcads in the Moscatel {Adoxa mosc/ialellina, L.) is often cited a.s an e.xaiuple of packing tlie nia.ximuin niimber of flowers in the niininium of Space. An inflorescence of the Moscatel gcnerally consists of five flowers — oiio ternuual and four lateral. Tlic terminal flower has two bracts, four petals, four branchcd st^imeiis and four carpels ; and each of the lateral flowers possesses three bracts, five petals, five branched stamens and five carpels. This is the most common form of arrangement, but the plant varies con- siderably and the following figiires show tliat only 55 per ceut. of the numbers connted agree with the above description, and 51 other methods of arrangement are noted. The number of flowers in the inflore-scence is a variable quantity as well as the number of parts in the periauth, and for the sake of convenience thcse variations are considered separatcly. The si)ecimens e.xamined were obtained from three localities, Chislehur.st in Kcnt, Catcrham in Surrey, and Theydon Garnon in Essex. I. Variation in the number of flowers in an inflorescence. The number of flowers constituting an inflorescence varies from three to tcn. 1071 in- florescences were counted and the following results were obtained : Mificedanea TABLE I. 109 Nuinber of flowcrs 3 Jf r> ''• 7 8 .'( 2 0-2 Actiial quantities... Percentages 6 0-6 71 6-6 934 87-2 26 2-4 24 21 7 0-6 1 Ol Diagram I. rcpreseiits thc above figuros in a grapliical form. 3 4 & 6 7 6 9 70 RO nn 40 30 «0 10 1 / 1 1 / DiAC.KASi I. Nuniber of flowers per inflorescence. In Gerard's Ihrhul (2iid Ed. 163.3), p. 1090, there is a nute relating to the Moscatel which is interesting. He say.s " the flouros grow clustering on tlie top of a stalk, commonly iive or seven togetlicr, ; it floure.s in Aprill and is to be found in divers places amongst bu.-ihea at that time, as in Kent about C'lii.selliurst, esi)ecially in Pits bis wood and at the fnrthor end of Cray heath on the left hand undcr a hedge among bryers and brainbles, which is liis proi)er seat." (icrard gives an Illustration of the Moscatel wliich sliows two Howering stcms, oiie 110 Miscellanea of these has a clustered group at thc top «ith one small lateral flower sonie distance down the stem, anJ the other a clustered group with two lateral flowers in a lower position. Several plants bearing seven flowers were found ncar the spot mentioned by Gerard as "Cray hoath." It seems stränge that Gerard sliould give the numbcrs as tivc to scven wlien the four-flowered inflorescences occur niore thaii twice as often as either the six-flowered er the seven-flowered. IL Vanation in the number of petuls. Before considering the Variation in the number of {Metals it would be as well to say a few words about the niode of arrangement of the parts of a flower of the normal type. The tips of each of the uppermost petals of the four pentaiuerous lateral flowers (1, Diagram II) fit into the space betweeii two i)etals (4) of the terminal and tetramerous flower. DiAOBAM II. Plan and side view of infiorescence without terminal flower. The arrangement of the reraaining petals is sliowii in Diagram II, where the pet;il indicated by 2' in the expanded lateral flower shown in the diagram touchea petiil 2 of the flower on the left band, while ])ctal 2 of flower A touches 2' of it-s neighbour on the right. The jietals marked 3 and 3' touch in a similar manner. The Order in which the various flowers and petals in an infiorescence geuerally ojwn is very curious. First, the four petals of the terminal flower e.\pand sinniltJvneously, theii two lateral buds which are dianietrically opposite (a and a') open, and finally the opening of the other pair of buds (b and b') take-s place. The order in which the petals of the pentamerous flowers exi>and is 1, 2 and 2', and 3 and 3'. Matheuiatical precision is nece.ssary where tlie flowers are packed in such a neat fa-shion, otherwise the struggle for space would result in a haphazard arrangement and consequently a loss of space. The terminal flower is the least variable of all. Out of 763 terminal Howers 754 (about 99 per cent.) were found to be tetramerous ; 4 being pentamerous and 5 trimerous. The fi.xity in the numl>er of parts of the terminal flower is used for the pur[X)se of classifying and tabulating the variations from the normal type as given in Table II. The numbcrs given in Table II. lines 16 — 20 deserve special iiotice. Line 16 shows that 420 inflorescences with 4 pentamerous lateral flowers (reprcsented by the entry 4 in column 5) were noted. In line 17 whero tliere are 3 "tivcs" and 1 "four" in thc inflorcscenco the numl>cr hiU4 droppcd to 125. In line 18 where there are 2 "fives" and 2 " fours'' the number is 59, and lines 19 and 20 show a further decrease in the totals as the florets with four jietals take the places of those with five. The same rule holds in cji-ses where the numbcrs of flowers in the inflorasccnces are seven, six and four. Miscellanea 111 TABLE II. VaHations in tlie Nmnher of Divisions of the Curulla. Terminal Flower Total Number of Flowers per in- florescence Distribution of Petals lateral flowers* in Locality Totals 1 K o S 8 G 5 Jf 3 Near Chislehurst, Kent Caterham, Surrey Theydon Garnon, Essex Tri- merous ■ 1* — 1 — 2 2 1 1 2 1 1 — 1 1 2 3 h — — — — 3 — — — 2 4 f 3 1 — — 1 — — 1 — — 5 3 — — — 2 — — 4 — — 6 It — — — 3 — — 16 3 2 21 7 k — — — 2 1 — 4 1 — 8 k — — — 1 2 — 3 — 1 9 J, — — — — 3 — — — 1 10 k — — 2 — 1 — 1 — — 11 k — — 1 2 — — 2 — — 12 J, — — 1 1 1 — 1 — — 13 k — — 1 — 2 — 1 — — 14 A — — — 2 — 1 1 — — 15 5 — — — 4 — — 283 30 107 420 16 5 — — — 3 1 — 91 6 28 125 17 5 — — — 2 2 — 38 10 11 59 18 5 — — — 1 3 — 27 1 4 32 19 5 — — — — 4 — 10 2 2 14 20 5 — — 3 — 1 — — 1 — 1 21 5 — 2 2 — — 1 1 — 2 22 5 — — 2 1 1 — 1 — — 1 23 5 — — 1 3 — — 4 3 — 7 24 Tetra- ^ merous " 5 5 5 — 1 1 2 1 3 1 2 1 3 1 1 1 1 1 5 2 1 25 26 27 5 — — — 2 1 1 1 — — 1 28 5 — — — 1 2 1 7 — — 7 29 5 — — — 1 1 2 — — 1 1 30 5 — — — — 3 1 2 — — 2 31 6 — — 1 4 — — 1 — 1 32 6 — — — 5 — — 5 — 1 6 33 6 — — — 4 1 — 1 — — 1 34 6 — — — 3 2 — 1 — — 1 35 6 — — — 2 3 — 1 — — 1 36 6 — — — 2 2 1 — 2 — 2 37 G — — — 1 4 — 2 2 4 38 i — — — 6 — — 5 — — 5 39 7 — — — 5 1 — 1 1 1 3 40 ~ — — — 4 2 — — 1 — 1 41 7 — — — 4 l 1 1 — — 1 42 7 — — — 3 3 — — 1 — 1 43 7 — — — 3 2 6 3 1 1 1 2 — — 1 1 2 44 45 46 7 3 8 — — — 6 1 — 1 — — 1 47 5 — — — 4 — — 1 — — 1 48 Penta- 5 — -^ — 2 2 — 1 — — 1 49 merous ' 5 — — — 1 2 1 1 — — 1 50 6 — — — 2 3 — — 1 — 1 51 * The figures at the top of this column denote tbe number of petals per flower. The figure.s in the vertical columns denote the number of flowers of each type. 112 Miscellanea It will be Seen that tbe corollas of the Ädoxa are 8-, 7-, 6-, 5-, 4- and 3-partite, and the actual number of cach of these types of corolla is shown in Table 111, while Diagram III gives the results in a graiihical form. Divigions of corollas. El 00 - 1 ^H 1 1 t- - 1 1— t 1 1 50 s 1 g 00 o 1 ■^ ■* « n 05 a> o ^ o 00 (N o a • f— t ■9 : : -h3 |. 1 : i s1 60 a . 2 U "2 ■ : 1 c u p1 s 5i |1 OQ 1 ■8*3 ^ 11 i'" 55 ^; -^ 3 4 6 e 1 1 1 1 1 1 1 1 1 1 1 1 60 t 1 1 , 1/ 1 / Percent 1 l 30 1 i \ 20 / / / f ( 1 / ( 10 / ( 1 $ 1 1 1 1 j 1 , a i 2 a * = s a ü"3. The Variation froni the normal type thercfore favoiirs the tetrameroiis type more than tlie pontamerous ty[io, while siiorts occur in tlie form of 3-, C-, 7-, aiid 8-partite corollas though these form less than 2 per cent. of the total. NOTE. At the snggestion of Prof. Karl Pearson sevei-al olJ herbaria at Kew, the British Museum and at {Cambridge were examined in order to see whether any of the speciniens agree with the figure given in Gerard's Berbcil, 2nd Ed., p. lOOl (edited by Thos. Johnson). None were found exhibiting the peculiar racemose arrangement which is so niarked in his drawing. The first edition (publislied 1597) has an entirely different figure of Adoxa, which has evidently been borrowed from the Kriiiitn- Buch of Tabcrniemontanus. In this case all the lateral flowers are tetramerous. I am indebted to Miss K. M. Hall, Curator of the Stepney Borough Museum, for niuch lielp and many useful suggestions. REFERENCES. Eichler. Blutendiagi-amme. 1875 — 1880. Engler and Prantl. Die Naturalichen Pflanzenfamilicn. 1891. GiLTAY. " Ueber Abnormitäten in der Bl. von A. mosfhatellina." Xederlandsch Kruidkundig Archief, 2 ser. in. 4, p. 434. Henslow, llev. tt. " Ou the Origin of Floral .Estivatious." Trans. Lin. Soc. (Botauy), 2nd ser., Vol. i. p. 194. Masters. Vegetable Teratology. (Ray Soc.) 18G9. Wtdler. "Morphologische Mitteilungen, Adoxa moschatellina." Bot. Zeitung, 1844, p. 657. IV. Seasonal Change in the Characters of Aster prenanthoides, Muhl. In the last number of Bioiuetrilcd (pp. .304 — 315) a colleetion of evidence was p\ibli.shed, showing the existence of periodic changes in tlie mean and modal numbers of floral organs, and in the constants expressing the Variation of these numbers, in several plants. Just before the pnblication of this article, Mr G. H. Shull published in the American Naturalist, Vol. .\xxvi. No. 422, an elaborate " Quantitative Study of Variation in the Bracts, Rays and Disk-florets "of Aster Shurtii Hook., A. Nocae AmjUae L., ^-1. puniceus L., and A. prenanthoides Muhl., "from Yellow Spring.s, Ohio." In this paper the constants expres.sing the Variation in and correlation between bracts, rays, and disk-florets are fidly determiued for each species ; and in addition, the author describes the results obtained by examining four sets of flowers, collected from the sanie series of individuals of ,4. prenaiithaides on four different days. The first of these four gatherings was made ou September 27, the last on October 8 ; a coniparison between Biometrikii ii 15 114 Miscellanea the fovir sets of results gives a reniarkatile and instnictive demonstration of the change which may occur in individiial i>lant cliaractcrs withiii a .short sjiace of time. The following table, abridged froin tlie fuller tables given by Mr ShuU, shows some fejvtures of the change. TABLE I. Yariatiun in Nuinber und S. 1). of Flural Uiyuiin on four different Days. September 27 (117 Capitula) September 30 ai:s Capitulu) Ootober 4 (139 Capitula) October 8 (176 Capitula) Mean No. Bracts ... S. D. of Bracts Mean Xo. Ilays S. D. ofRays Mean N'o. Disk-tlorets S. D. Disk-tioi-ets ... 47-410 + 0-345 5-524 ±0-244 30-7(i9± 0-249 3-986 + 0-176 56-427 ±0-249 3-986 ±01 76 44-343 + 0-291 5-152 + 0-205 28 706 ±0-201 3-.569 + 0-142 51-713±0282 4-995±0-199 43-8:j5± 0-302 5-276±0-2l:5 28-252 + 0-200 3-501 ±0-142 491 58 ±0-279 4-885±0-198 41 -920 ±0-249 4-890 ±0-1 76 •26-335 ±0-1.53 3010 + 0-108 45-778 ±0-242 4-777 ±0-171 The change in mean and -Standard deviation of all the organs studied during the short interval between Scjitoinlxir 27 and September 30 is of especial intcrest in connection with the asserted miiltimodal distribiitions so often describcd in tloral organs. Mr Shull lays little stre.ss on the peaks of his frequency curves, but it i.s worth notice that in this carefully collected inaterial there is not a Single c ■J , L« 10 1 9 9 1 9 9 1 8 in 1 10 10 1 8 10 1 10 1 2 1 4 1 3 1 14 2 1 :i 1 (b) Of the formula " | " "^ ^ , 32 . .r 12s''. »7t m 7 8 8 8 7|9 8 |8 8 1 9 8 1 8 71 8 9 1 9 8 1 9 9 I 9 8 1 10 9 1 9 9 1 10 9 1 9 8 1 9 10 1 10 8 1 10 10 1 10 9 1 10 10 1 10 9 1 11 10 1 10 1 1 1 2 17 2 4 1 1 1 1 (c) Of the formula " , " , 26 or U)-4 7„ 8 1 8 8|7 7|7 9 |8 8 1 8 9| 8 9| 9 9 |8 8 1 8 10 1 8 8 1 8 lO'l 9 9 1 9 10 1 9 10 1 10 10 1 9 12 1 9 3 1 4 6 1 4 4 2 1 L. W. BVRNK 117 (W) OfMi,. fonnula " " , , 21 or 8-4. 7 III III. + k '" 7|7 7|8 8 1 8 7|9 9 9 7 9 8 1 8 9 1 9 8 1 9 8 1 8 8 1 10 9 1 9 8 1 8 9 1 9 9 1 10 9 1 9 10 1 11 8 1 9 8 1 10 9 1 10 1 1 1 5 6 1 1 1 3 1 3. Ainoug the 79 asyininctric;il in both serics tlie lollowiiig anaiiyeiiicnts occur: — (i) W'Iu'io botli .series have more plates on the same sidc, 40 (ir 10 '7^,. (a) Of the tbrmula rA — , 18 or 7'2 "/ . 8|7 8|7 9 1 8 9 1 8 9 1 8 10 1 8 9 1 8 10 1 9 9 1 8 10 1 9 10 1 7 9 1 7 10 1 8 9 1 8 10 1 8 9 |8 10 1 9 10 1 9 10 1 9 9| 7 1 2 2 1 1 1 1 1 7 1 (b) Of the fonnula " j " "*" f , 22 or 8-8 7„. ^ ' m m + l '° 7|8 7|9 8| 9 7|9 7 1 8 8 1 9 8 1 9 8 9 8 1 9 8 1 10 8 1 9 9 1 10 10 1 11 9 1 10 9 1 10 1 9 1 10 8 1 9 9 1 11 1 1 5 6 1 4 2 1(1 ' 1 (ii) Whore the phites of one siile jin'jioiMli'rate in one seriös anil of the otlier in the other. («) Where the total number of phites on the right side o(iiials the total nuniber on the left and the specimen attains a " secondary symmetr}'," 32 or 12-8 7.. )( + k n (1) Of the fornuila 111 III + 7;. 13 or 5-2 7, 8 17 9 1 8 8 1 9 8 1 9 10 1 9 8 i 9 9| 7 8 ! 10 9 1 8 10 1 9 9 1 10 9 1 10 9 1 7 9 1 11 1 5 1 1 3 j 1 ' 1 (2) ( )f the formula 11 I n + k in + k I 111 1!) or 7-0 7„. 7|8 9 1 8 8 1 9 9 1 8 7 |8 10 1 9 8 1 9 10 1 9 9 1 10 10 1 9 8 1 10 11 1 9 1 13 1 2 1 1 118 Bony riates of John Dory (b) Whcro tho pliitos »f ono side or the other prepoudcrato thcre arc 7 or 2"8% of the following furmula; and airangeinents : — n + lk 1 n m 1 m+k+l n + k+l 1 71 m 1 »71 + * n 1 n + i-+l m + t 1 m 9 1 8 9| 10 8 1 7 8 1 10 9|7 8 1 9 7|9 9 1 » 8| 10 9 1 8 8 1 10 10 1 11 1 1 1 2 1 1 The following are the particular combinations which occiir in moic thaii 5 per Cent, of the 250 specimens : — 8 I 8 9 I 9 in 2-5 specimens or 10 per Cent of the total. 8 9 . 9"' 17 „ if 6-8 9 9 8 . 14 „ » ÖG 9 9 . 14 „ t) 5-6 9 9 '" 8 9 . -0 "1 IS „ I) 52 The niimber of the plates. Dr Günther in the first edition of the British Museum Catalogue gives the total number of plates in Z. faher as 98, or 7 occa.sionally 10 in each dorsal scrie.s and 9 in each anal. In the specimens under consideration the maxinia and minima observed were : — Dorsal L. Max. 10 Min. 7 R. Max. 11 Min. 7 Anal „ „ 12 „ 7 „ „ 11 „ 7 , ,. ... , lU-7 I 7 - 11 and the Variation niay be diagraminaticaliy expressed z-^ — - . _ — ry . , r , «, ^ «432 I 8-464 , , The average number of plates was 34'6, s Sin ' 'argest in any .,..,, ,^ . 10 I 11 ,9 iiiiiiviniial 40, in two in.stances, -— r and j^JJ,andtheMnallest29,;|^. The accompanying table and diagram show the number of plates occurring in the in 36 — 3 — U — 27 2 1 7 1 1 l 3 — 4 — 4 1 — 1 5 2 1 2 — 3 1 5 1 6 — 1 — 6 — 1 — 14 — — 4 — 15 17 10 7 2 1 15 2 2 1 9 4 ÄS Ä.9 ^/O TDtal ! Per c«nt. 13 19 7 59 23C 33 13-2 32 12-8 26 10-4 21 8-4 18 7-2 22 8-8 7-G 2-8 Total 1 3 9 23 24 60 54 40 17 11 Per Cent. 3-6 9-2 9-6 24 21-6 16 6-8 4-4 2-4 \ 30 31 32 33 34 t^.. V, ~'*^^■&^-\ 1 " ^ . ^ ■^■-••. ^ \- 1. 1 '^j* \ ?^*^ — —^ 33 34 36 30 37 38 -y^ !^^ , '•" ' y.'.V-^^"-" ' — 86 80 37 3S 39 ./..-^•^ P' 1. Total 7i I n m I m 2. Asymmetrical in one series* 3. (i) Asymmetrical in both f — • 3. (ii) Secondarily symmetrical * Very few specimens bave an even number of plates and these are omitted in traciug tlie diagram. + Specimens witb an odd number of platea omitted. 120 BoHji Plates of John Donj So far as can be judgcil from such a sni.ill iiuiiil)ir of specimens thc foUowing points seem worth iidtiiig: — 1. The plates are arraiigeil syiniiietrically in mily li.Sö per eent. of the whole. 2. The average niimber of plates is 340, am! thc actuiil uuinbers of iiiost frcqueiit ocourrence 34, 3."), and 36 (oiie of these three oucunlng in over 60 per Cent, of the total). 8 I 8 3. The most frequent combination is ^^-t-^ which occurs in 10 per cent. of J I .) the total; the other most iisnal conibinations aro those in wliieti one or both of the uppcr series are increased to 9. i Ti 1 f K f I f . ] in 10 I 11 , 9 I 11 , 4. Ihe largest iiuniber oi plates iioted was 40, - ~ and ^, aiul 7 I 7 the smallest 29, ^ „ . 7 ö Biometrika, Vol. II. Part Plate I. ^'cmll'4 .li>lin Diirv, showiiii; |insiti"ii (if .-MitiM'inr dor.s.al aiul ;ui;il [ilntos. CD CO QQ CD r — C3 I — C3' will Multiply, Divide, Add, Subtraot, and consequently will perform all nnmerioal complications in piain figures or decimals. Invaluable for compilin;; Statistics, working out Ratios, Averages, Percentages, Birtb Katea, Death Eates, Interests, Foreign Exchanges, &c., &c. Size A. Products 18 Figures ; Multiplier or Quotient 10 Figures. £Z1. lOs. Size B (the most populär). Products 13 Figures ; Multiplier or Quotient 8 Figures. £20. Size C. Products 10 Figures ; Multiplier or Quotient 8 Figures. il5. Agents— T, WHITTINGHAM & CO., Limited, Export Stationers and Printers, 35, BUCKLERSBURY, LONDON, E.G. Registered Office and Works-12, LITTLE TRINITY LANE. JOURNAL OF THE ANTHROPOLOQICAL INSTITUTE. Vol. XXXII. Pts. 3 and 4. 1902. Among the Contents are : — The Third Annual Huxley Memorial Lecture : Eight-Handedness and Left-Brainedness, by Prof. D. J. CuNKiNGHAM, M.D., F.B.S. Tho Medicine and Surgery of the Sinaugolo, by C. G. Seligmann, M.D. Notes on the Wagogo of German East Africa, by Eev. H. Cole, C.M.S. On the Classification and Arrangement of the Material of an Anthropological Museum, by W. H. Holmes, Chief of the American Bureau of Ethnology. On the Ethnography of the Nagas, by W. H. Fdkness, M.A., M.D. Some Pre- liminary Eesults of an Expedition to the Malay Peninsula, by N. Annandale and H. C. Robinson. Skulls from Botuma, by W. L. H. Ddckworth, M.A. On the Religious Beliefs of the tribes of the Papuan Gulf, by the Rev. J. Holmes. Du the Initiation Ceremonies of the Natives of the Papuan Gulf, by the Eev. J. Holmes. The Ceramic Type of the Eaily Bronze Age in Britain, by the Hon. J. Aber- OBOMBY. Anthropological Notes on Southern Persia, by Major P. Moleswobth Stkes, C.M.G. Etc., etc. WITH TWENTY FDLL-PAGE PLATES. NUMEROÜS ILLUSTRATIONS IN TEXT. EEADY IN JANUAEY. Published by THE ANTHROPOLOGICAL INSTITUTE of GREAT BRITAIN and IRELAND, 3, Hanover Square, London, W. And to be obtained through all Bookseilers. MAN. A MONTHLY RECORD OF ANTHROPOLOGICAL SCIENCE. Published uuder the direotion of the Anthropological Institute of Great Britain and Ireland. Each number of MAN consists of 16 Imp. ovo. pages, with illustrations in the text together with one fuU-page plate ; and includes Original Articles, Notes, and Correspondence ; Reviews and Summaries ; Reports of Meetings ; and Descriptive Notices of the Acquisitions of Museums and Private Collections. Prioe, 1«. Monthly or lOi. per Annum prepaid. TO BE OBTAINED FBOM THE ANTHROPOLOGICAL INSTITUTE, 3, Hanover Square, London, W. AND THBOUGH ALL BOOKSELLEBS. CONTENTS {AU Rights reaerved) I. On thc Systciuatic Fitting of Curves to Observatioiis and Measurements. Part IL (With 8 Figures.) By Karl Pearson, F.R.S. . . . 1 II. Quantitative Study of the Effect of Eiivironraeut upou the Fomis of Nassa obsoleta and Nussa trivittata from Cold Spring Harbor, Long Island. (With 5 Figures.) By Abigail Camp Dimon ... 24 III. On the Ambiguity of Mendels Categories. By W. F. R. Weldon, F.R.S 44 IV. Cooperative Investigations on Plauts : I. On Inheritance in the Shü-ley Poppy 56 Miscellanea. (i) Note on the Results of Cros-sing Japanese Waltzing Mice with European Albino Races. (With 4 Figures.) By A. D. Dakbishire 101 (ii) Interpolation by Finite Differcnees. (Two Independent Variables.) By W. Palin Elderton .... 105 (iii) Variation in. the Möscatel {Adoxa Moschatellina, L.) (With .3 Figures.) By Henry Whitehead . 108 (iv) Seasonal Change in the Characters of Aster prenantlundes (Muhl). Note ou a paper by G. H. SriULL . . 113 V. On the Number and Aa-rangemeut of the Bony Plates of the Young John Dory. By L. W. Byrne. (With one Figure and Plate I.) . 115 The Editors wish to acknowledge the receipt of contributions from W. R. Macdonell, Norman Blanchard, Alice Lee, Frank Lutz, L. Doxcaster and E. H. J. Schuster, and E. IJ. J. Schuster. Biometrika will appear about four times a year, A volutne containing about 400 pages, with plates and tables, will be issued annually. Papers for publication should be sent to Professor W. F. R. Weldon, Merton Lea, Oxford, to Professor Karl Pearson, 7, Well Road, Hampstead, London, or to Professor C. B. Dav^nport, University of Chicago, Illinois, U.S. It is very desirablc that a copy of all measurements made, not uecessarily for publication, should accompany each paper. 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Reynolds Green, Sc.D., F.R.S., Professor of Botany to the Pharmaceutical Society of Great Britain. Second Edition. Demy 8vo. 12». Practical Morbid Anatomy. By H. D. Rolleston, M.D., F.lt.C.P., and the late A. A. Kanthack, M.D., M.R.C.P. Crown 8vo. 6». LONDON: C. J. CLAV d SONS, CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, AVE MARIA LANE. Volume II FEBRUAIIY, 1903 No. 2 NOTES ÜN THE THEORY OF ASSOCIATION OF ATTRIBUTES IN STATISTICS. By G. UDNY YULE. 1 ... >.1>Y CONTENTS. BOT.^NICAL GARDEN Page IntrodiR'tory ............. 121 1. Notation ; terminology ; tabulation, etc 122 2. Consistence and iuference 124 3. Association 125 4. On the theory of coraplete independence of a seriea of Attributes . . 127 5. On tlie fallacies that may be caused by the mixing of distinct records . 132 The simplest possiblo form of Statistical Classification is " division " (as the logicians term it) " by dichotomy," i.e. the sorting of the objeets or individuals observed into one or other of two mutually exclusive classes according as they do or do not possess some character or attribiite ; as one may divide meii into sane and insane, the members of a species of plauts into hairy and glabrous, or the members of a race of animals into males and female.s. The raere fact that we do eniploy such a Classification in any case must not of course be held to imply a natural and clearly defined boundarybetween the two classes; e.g. sanity and insanity, hairiness and glabrousness, may pass into each other by such fine gradations that judgments may differ as to the class in which a given individual should be entered. The judgment must however be finally decisive; intermediates not being classed as such even when ob.served. The theory of statistics of this kind is of a good deal of importance, not merely because they are of a fairly common type — the statistics of hybridisation experiments given by the followers of Mendel may be cited as recent examples — but because the ideas and conceptions required in such theory form a useful introduction to the more complex and less purely logical theory of variables. The classical writings on the subject are those of De Morgan*, Boole f and JevonsJ, the method and notation of the latter being used in the following Notes, the first three sections of which are an abstract of the two memoirs referred to below§. * Formal Logic, ehap. vm., " Du the Numerically Definite Syllogism," 1847. t Amdysis of Logic, 1847. Laics of Thought, 18.54. t " On a General System of Numerically Definite Reasoning," Memoirs of Manchester Literanj and Philosophical Society, 1870. Eeprinted in Pure Logic and other Minor iVorks, Macmillan, 1890. § "On the Association of Attributes in Statistics," Phil. Trans. A, Vol. 194 (1000), p. 2.57. " On the theory of Consistence of Logical Class Freijuencies," Phil. Trans. A, Vol. 197 (1901), p. 91. Biometrika ii 16 122 Oh the Theorij of Association 1. Notation ; terminology ; relations between the class frequencies ; tahulation. The Dotation used is as foUows * : N = total number of observations, {A) = no. of objects or individuals possessing attribiitc A, (a) = „ „ not possessing attribute A, {AB) = „ „ possessing both attributes A and B, {Aß) = „ „ „ attribute A btit not B. {aB) = „ „ „ attribute B but not A, {aß) = „ „ not possessing either attribute A or B, and so on for as many attributes as are specified. A class specified by n attributes in this notation may be termed a class of the «th order. The attributes denoted by English capitals may be termed positive attributes, and their contraHes, denoted hy the Greek letters, negative attributes. If two classes are such that every attribute in the one is the negative or contrary of the corresponding attribute in the other they may be tcnucd contrary classes, and their frequencies contrary frequencies; {AB) aud {aß), {ABy) aud {aßC) are for instance pairs of contraries. If the complete series of frequencies arrivcd at by notiug n attributes is being tabulated, frequencies of the same order should be kept together. Those of the same order are best arranged by taking separately the set or " aggregate " of frequencies, derivable from each positive class by substituting negatives for one or more of the positive attributes. Thus the frequencies for the case of three attributes may conveniently be tabulated in the order — Order 0. iV Order 1. {A), {a) : {B),{ß): {G), {y) Order 2. {AB). {Aß), {(^) as {Aß) > {B) a.s {aB) (4); and similar conditions inu.st hold for (AC) and {BC). But these are not the ouly conditions that nuist hold. The second-ordor frequencies inust not only be such as not to imply negative values for tlic frequencies of other classes of their own aggregates, but also must not iinpli/ negative values for any of the third-vrder frequencies. Expanding all the third-order frequencies in terms of the frequencies of positive classes, and puttiug the resulting expansion {AB) >(AC) >{BC) > {AB) + {AC) + {BG) - {A) - {B) - (0) + X or the freqnency given below will be negative {ABC) [l]] {Aßy) [2] {aBy) [3] {aßC) [4] {ABy) [ö] {AßC) [ü] iaBC) [7] {<^ßy) [«] (5). But if any one of the minor limit.s [1] — [4] be greater than any one of the major limits [.5] — [8] these conditions are impossible of fulfilment. There are four minor limits to be compared with four major limits or sixteen comparisons in all to be made ; but the majority of these, twelve in all, ouly lead back to conditions of the form (4). The four comparisons of expansions due to contrary frequencies alono lead to new conditions — viz. {AB) + {AC) + {BC) ^{A) + {B) + (C) - N {AB) + {AC)-{BC)1f>{A) {AB)-{AC) + {BG)1^{B) -{AB) + {AC) + {BC)::^{0 (6). : G. U. YULE 125 These conditions give liinits to aiiy ouc of thr t-hree frequencies (j4ß), (/16') aiid {BC) in terms of the other two and Ihc fnciui'iicii's of tlu; first order, i.e. enablc us to infer liniits to the one class-frcinu'ncy in terms of the othors. It will very usually happen in practica! Statistical oases that the limits so obtaincd are vahie- less, lying ontside those given by tlie simpler conditions (4), but that is merely because in practice the values of the assigned fretpiencies, e.g. (AB) and (AC), seldom approach sutticiently closely to their limiting values to render inference possible. 3. Association. Two attributes, A and B, are usually dcfinod to be indepcndent, witliin any given field of Observation or " universe," when the chance of finding theni together is the product of the chances of finding either of them separately. The physical meaning of the definition seems rather clearer in a different form of Statement, viz. if we define A and B to be independent ivhen the proportion of A's amongst the B's of the yiven universe is the sarne as in that universe at (arge. If for instance the (piestion were put " Wliat is the test for independence of sniall-pox attack and vaccination ?", the natural reply would be " The percentage of vaccinatetl amongst the attacked should he the same as in the general popu- lation " or " The percentage of attacked amongst the vaccinated should be the same as in the general population." The two definitions are of course identical in efFect, and permit of the same simple symbolical expression in our notation ; the criterion of independence of A and B is in fixet (AByJ'^P ,7). In this etpuition the attributes spocifying the universe are understood, not expressed. If all objects or individuals in the universe are to possess an attribute or series of attributes K it may be written (AK)(BK) (ABA) = ^j^y-. An equation of such form must be recognised as the criterion of independence for A and B within the universe K. As I have shewn in the finst memoir referred to in note §, p. 121, if the relation (7) hold good, the three similar relations for the reniaining fretpiencies of the " aggregate " — i.e. the set of frequencies obtained by substituting their contraries a, ß for A or B or both — must also hold, viz. (A)(ß)\ •(8). 12ü On the Theory of Association for we have at once ainl so Uli. TliL' case ot two attributes is tlius quite simple; the definition of iiidependence is almost intuitive, and the criterion need only be applied to one frequenc}' of tlie aggregate. The two attributes arc termed positively or uegativeiy associated accordiiig ;us {AB) is greater or less than the value it would have in the case of independeuce, or, to put the same thing aiiother way, accordiiig as (AB)j{A) is greater or less than {B)IN or {AB)/(B) greater or less than {A)IN. If iiiorc than two attributes are noted in the record they must be dealt with, in the lirst instance, pair by pair a-s above, but subsequentiy the assoeiation between each pair should be observed in the sections of niateiial or sub-universes defined by the other attributes. In the case of three attributes, for instance, we have not only to deal with the assoeiation between A and B in the universe at large but also in the universe of Cs and in the universe of 7's — associatiuns wliich will be tested by comparing {ABG)I(AG) with (BC)l{C) or {ABC);iBC) with (AC)l(C), and (ABy)j(Ay) with (ß7)/(7) and so on. Such "partial" associations are of great practical importance, as a test for the correctncss or otherwise of physical interpretations placed on any " total " associations observed. When A and B are found to be associated it is a common form of argument to say that the assoeiation is not " direct " but due to the assoeiation of A with G and B with 0; the argument may be tested at once by finding whether A and B are still associated in the separated universes C and 7. If they are the argument is baseless. It has been said for instance that the assoeiation between vaccination and protection from small-po-x is due to the assoeiation of both with sanitary conditions (a larger proportion of the upper classes than of the lower classes being vaccinated, accordiug to the argument). To test the argument the ''universe" or material observed should be limited eitker to tliose liviug wholly in .sanitary conditions or to those living wholly in insanitary conditions (it does not matter which). 'Partial" associations again arc of importance to the biologist in the theory of heredity. If an attribute be hcritable, its presence in the parent or the grandparent is associated with its presence in the offspring; but the physical Interpretation to be placed on such inheritauce depends very largely on whether there is also a partial inheritance from the grandparent, the presence of the attribute in the grandparent being associated with its presence in the offspring even when the parents either all possess or all do not possess the attribute. It is important to notice that the test for assoeiation is necessarily based on a comparison of perceiitagcs or proportions, e.g. {AB)'{A) and {B)IN. The mere fact that a certain number of A'a are B gives no physical Information; besidcs knowing how many A'^ are B you must kiiow also how niaiiy non-.4's are B or what proportion of A's e.xists in the given universe at large. In an investigation as to the causation of A it is therefore just iis important to observe non-.4's as A'a. G. U. YüLK 127 This poiiit is t'requently t'orgottcn. In an iuvestigation as to Uiu iiiheritance of dcaf-mutism in America*, for instance, only the offspring of deaf-mutes were obscrvcd, and the argument consequently brcaks down on page after page into conjectural statcmcnts as to points on which the editor has no inforniation — e.g. the Proportion of deaf-mutes amongst the children of normals. The difference of iAB)/{A) frcia (yi)/.Y and of {AB)!{B) from {A)!N are of coursc not, as a ruie, the same, and it would be useful and convenient to measure the " association " by some more symmetrical method — a " coefficient of association " ranging between + 1 like the coefficient of correlation. In the first memoir referred to in note §, p. 121, such a coefficient, of empirical form, was suggested, but that portion of the memoir should now lio read in connection with a later memoir by Professor Pearson f. 4. On the theory of complete iiidependence of a series vf Attrilmtes. The tests for independence are by no means simple when the inimber of attributes is more than two. Under what cirenmstances should we say that a series of attributes ABCD... were completely independent ? I believe not a few statisticians would reply at once " if the chance of finding them together were equal to the product of the chances of finding them separately," yd such a reply would be in error. The mere result {ABCD...) JA) (B) (C) (D) N N ' N ' N ' JSr ^■' does not in general give auy informatiou as to the independence or otherwise of the attributes concerned. If the attributes are knoiun to he completely inde- pendent then certainly the relation (0) holds good, but the converse is not true. " Equations of independence " of the form (9) must be shewn to hold for more than one class of any aggregate, of an order higher than the second, before the complete independence of the attributes can be inferred. From the physical point of view complete independence can only be said to subsist for a series of attributes ABCD... within a given universe, when every pair of such attributes exhibits independence not only within the universe at large but also in every sub-universe specified by one or more of the remaining attributes of the series, or their cnntraries. Thus three attributes A, B, C are completely independent within a given universe if AB, ^t'and BC are independent within that universe and also AB independent within the universes C and 7, -^C' „ „ „ B „ ß, ^G „ „ „ A „ a. * Mnrridflfs of the Deaf iti Amerirn, ed. by E. A. Fay. Volta Bureau, Washington, 1898. t Plul. Trans. Vol. 195, p. 16. 128 On (he Theory of Association If a series of attributes are complctely imlependent accordiiig to this definition relations of thc form (9) must hold for the frequency of every class of every possible Order. Take the class-frequency (ABCD) of the foiirth order for instauce. A and B are, by the terms of the definition, independent witliiii thc universe CD. Therefore (ABCD)J-^(^\ {LI)) But A and 0, and also B and 0, are independent within the universe D. Therefore the fraction ou the right is equal to 1 (AD) (CD) (BD) (CD) _ {AD)(BlJ) {CD) (CD)- {D) ■ (D) (Df But again AD, BD, CD are cach independent within the univeree at large; therefore finally (Anrm- ^ (A)(D) {B){D) {G)(D) {A){B){C){D ) {ABtD)--^^.^^~ .^-^-.-^= W • Any other frequency can be reduced step by step in preci.sely the same way. Now consider the converse problem. The total freqni'ncy N is given and also the n frequencies {A), (B), (0), etc. In how many of the ultimate frequencies {ABCD. ..MX), {aBCD...MX), etc. must "relations of independence" of the fonn {A){B){C){D)...{M){X) {ABCD...MN). m- hold good, in order that complete independence of the attributes may be inferred ? The answer is suggestcd at once by the following consideration. The number of ultimate frequencies (frequencies of order n) is 2"; the number of frequencies given is /i + 1. If thcn all but n + 1 of the ultimate frequencies are given in terms of the equations of independence, the remaining frequencies are deter- minate ; either thcse detcrminate values must be those that would be given by equations of independence, or a State of complete independence must be impossible. Supposc all the ultimate class-frequencies to have been tested and found to be given by the e<)uati()ns of independence, with the exception of thc negative class ( 5/378... /ii«) and the n classes with one positive attribute {AßyB . . . /jlv), (aByo . . . nv), etc. Take any one of these untested class-frequencies, {AßyS ... nv), and we have for example iAßyB...^iv) = {A)-{ABCD ... MN) -{ABCD...Mv) — other terms with one negative -{ABCD ... fxv) — other terms with two negatives — {AByB ... fiv) — other terms with w — 2 negatives. G. U. YULE 129 Bat all tlio f'reiiuencies ou tho right are giveii hy tlie rclations of iadependence. Thereforc {AßyB...^u)=^/^^ (iV- -(/i)(C)(Z))...(Ji)(i\0 -(B){C)(D)...{M){v} -iB)(C)(D)...(f.)(v) -(B)iy)iS)...(f^){v) - )• Now consicler the terms on the right in tho bracket. With the exception of the one tena (ß) (y) {S)...{fj,){i>), the remainder can be grouped in pairs of which the one moiaber contains (B) and the other (ß), the following frequencies in each niember of the pair being the sarne. Carrying out this rearrangement the expression will read iAßyS...i,v)=jß-^ jiV«-'-(i?)(.y)(8)...(M)W -iB){C){D)...{M){N) -{ß){C)(D)...{M){N) -{B){C){B)...(M){v) -{ß)(C){D}...{M)(v) -(B){C)i8)...if.)(v) -(ß)iC){8)...(f.)(v) - )• Replace (B) by N - (ß) throaghoiit and rearrange the terms in similar pairs containing C and 7. (B) and (ß) are then eliminated from all the terms and the expression then becomes {AßyS...f.v) = ^_- {iV- + (ß) (y) (8) . . . (f.) (v) -N(y){S)...(^)iv) -N(G)(S)...(f.){v) _ ,_ -N(C)(D)...(M)(N) -N{y)iD)...(3r,(N) -N{C){D)...{M){v) -Niy)iD)...{M)(v) - !• Biometiika 11 17 130 Oll the Theonj of Association Replaciiifj C by N - {-y) aiul regrouping in similar pairs of terms containing (i)) and (8) this will become {Aß^Z ...H-^) = jß, {i\^»-' + (ß) (7) (8) . . . (^) («') -i\r'(Z)) (£)... (J/)(iyr) -i\r»(8)(^)...(j/)(ir) — etc.) and contimiing the sanie process until .all the frequencies (D) (E) . . . (^f) (X) are eliminated, i.e. n — 1 times altogether, (A)(ß)(y)(8)...{t.){v) (AßyB ... fiv) = iV- That is to say the theorcm niust be triie quite generally : " A serie.s of n attributes ABC ... MN are completely independent if the relation.s of iiidependence are proved to hold for (2"— n+l) of the 2" ultimate frequeucies; such relations niust then hold for the remaiuing n+\ frequencies also." If the ultimate frequencies are only given by the relations of independence in n cases or less, independence may ex ist for certain pairs of attributes in certain universes but not in general. The mere fact of the relation hoiding for one cla.ss, e.g. iABGD . . . MiV) = (^)(^)(^ 'K^^- • •_ WW , implies nothing — in striking contrast to the simple case of two attributes, where 2" — jt+ 1 = 1 and ouly the one class-frequency need be tested in order to see if independence exists. In the case of three attributes the number of third-order classes is eight, of which four must be tested in order to be certain that complete independence exists. In the case of four attributes there are sixteen fourth-order classes of which eleven must be tested, and so on. I have dealt with the problem hitherto on the assumption that only the first- order and the 9(tii order frequencies were given, and that the fiequencies of intermediate onlcis were unknown— or at least uncalcuiated, for of course the frequencies of all lower Orders may be expressed in terms of those of the »ith Order. If however the fre((uencies of all onlers may be supposed known, the above result may be thrown iuto a somewhat interesting form. It will be remembered that the frequency of any class of any order may be expressed in terms of the frequencies of the positive classes [(A) {AB) (AG) (ABC) etc.] of its own and lower Orders. Tlien complete independence exists for a series of attributes if the criterion of independence hold for all the positive-class frequencies up to that of the nth Order. If \ve have for instance {ABCD ... MN) = ^^_^ {iA){B){C)iI)) ... W(iV)}, G. U. YuLE 131 and also (ßCD ... MN) = ^— m{('){D) ... (il/)(iV)l, \ve niu«t havc (aBCD . . . MN) = {BCD . . . MN) - {ABCD . . . MN) = ^. m{C){D) ... {M){N)\ [N-{A)] = ^M{B){C){D)...{M){N), and so od. The nuniber of clas.s-frequencie.s to be te.stcd in ordcr to demonstrate the existence of cunipletc independeuce is, of cour.se, the same as bofore, viz. It should be noted as a consequence of these results that the definition of "complete independence " given on p. 127 i.s redundant in its terms. It is quite true that if complete independence subsist for a series of attributes every possible pair must exhibit independeuce in every possible sub-universe as well as in the iiniverse at large, but it is not necessary to apply the criterion of independeuce to all these possible cases. In the case of three attributes for instance the criterion of independence need only be applied to four frequencies, as \ve have just seen, in Order to demonstrate complete independeuce ; it cannot then be necessary, as suggested by the definition, to test niue different associations, viz. I ^5 I \ AB\G\ \AC\ \AC\B\ \ BC \ I 56' I 4 I in the uotation of my memoir on Association (au expression like | .45 | G \ specifying " the association between A and B in the universe of (7's"). It is in fact only necessary to test |-45|, I^CI, \BG\, and AB\G\ (or one of the other three partial associations in positive universes). If these are zero, the remaining associations must be zero also ; for we are given AB\ i 7 AG 1/3 BG ^ (ABG)^-^^ {AG){BG) = ^.^ (A) (B) (G), ■.(ABG) = ^^(AB)iBG) = ^i^(^5)(46') = etc. i.e. I AG\ B\,\ BG \ A |, etc. are zero. Quite generally, it is ouly necessary, if the testing be suppcsed to proceed fmm the second order classes upwards, to test one of all the possible partial associations corresponding to each positive class. If there be four attributes ABGD, the six total associations | AB |, | AG\, \ AD \,\ BG \ 17-2 132 On the Theonj of Association etc. miist first be tried ; if thcy are zero, then follow on with | AB \C\,\ AB \D\, ! AC\ D 1 and \BG\ D\, but not . AC\ B \ or \ AD j B \ etc. ; if they arc zero then finally try j AB \ CD \, if it also be zero tlien the attributes are completely inde- pendent. It is not necessary to try further | AG\ BD \ or | AD \ BC \ etc. The inadequacy of the usual treatment of indepondence ariscs from the fact that it proceeds wholly d priori, and gcnerally has reference solely to cases of artificial chancc. The rcsult is an ilhisoiy appearance of siiiiplicity. It is pointed out that if ono "evcnt" can "succeod" in «i and " fail " in ^, ways, a second succeed in cu and fail in h^ ways, and so on, the combined evonts can take place (succeed or fail) in («i + ^)(a, + L)...(a„ + 6„) ways and succeed in ttlf/o ... «„ ways. The chance of entire "success" is therefore OiCfcj ... On (ai + 6i)(a, + fe,)...(((„ + 6„)' the chance of the first event failing and the rest succeeding is hiür, ... a„ and so on for all other possible cases. In short the chance of occurrence of the combined indepcudent events is the product of the chances of the separate events. There the treatment stops, and all practical difficulties are avoided. In such text-book treatment it is given that the events are independont and required to deduce the conscquences ; in the problems that the statistician has to handle the consequences — the bare facts — are given and it is required to find whcther the "events" or attributes are independent, wholly or in part. 5. On the fallacies that maij he caused hy the mixiiig uf distinct records. It foUows from the preceding work that \ve cannot infer indcpendence of a pair of attributes within a sub-universe from the fact of indepondence within the universe at largo. From \ AB \ = 0, we cannot infer | AB j C | = or | AB | 7 | = 0, although we can of course make the corresponding inference in the case of complete association — i.e. from | AB | = 1 we do infer \ AB\G\ = \ AB 1 7 [ = etc. = 1. But the convcrse theorem is also true ; a pair of attributes does not neces.sarily exhibit indepondence within the universe at large even if it cxhibit indepondence in every sub-universe ; given |.4B|C| = 0, 1 .45 1 7 1 = 0, we cannot infer |.45| = 0. Thr thooroni is of considerable practical importance from its invcrse application ; i.e. cven if |.(1B| have a sensible positive or G. U. YuLB 133 negative value wc cannot be sure that iicverthcless j AB \ C\ and j AB \ 7 | arc not bolh zero. Some given attribute might, fbr iiistance, be inherited neither in the male line nor the fcmale iine ; yet a niixed reeord might exhibit a considerabic apparent iuheritance. Suppose for instance that 50 "/o oi' the fathers and of tho sons exhibit the attribute, but only 10 % of the mothers and daiighters. Thcn if thcre be no iuheritauce in either line of descent the reeord must givo (approximately) fathers with attribute and sons with attribute 25 °/^ „ „ „ without „ 25 7o „ without „ „ with „ 25 % without „ 25% mothers with attribute and daughters with attribute 1 7„ , „ „ „ „ without „ 9 °/o „ without „ „ „ with „ 9 % „ without ,, 81 %. If these two records be mixed in equal proportions we get parcnts with attribute and ofifspring with attribute 13°/^ „ „ „ ,. „ without „ 17% „ without „ „ „ with „ 17°/„ „ „ „ „ „ without „ 53% Here 13/30 = 43^ "/„ of the offspring of parents with the attribute possess the attribute thcmselves. but only 30°/_ of offspring in general, i.e. thcre is quite a large but illusory iuheritance created simply by the mixture of the two distinct records. A similar illusory association, that is to say an association to which the most obvious physical meaning must not be assigued, may very probably oceur in any other case in which different records are pooled together or in which only one reeord is made of a lot of heterogeneous material. Consider the case quite generally. Given that j J.5 | ] and | ^5 | 7 | are both zero, find the value oi{AB). From tlie data we have at once tAR^\- (^^)(-^'y) _ [(Ä)-(AC)][ { B)-(BC)] ^^SC)J^C)(BC) Addins {AB) = (C) • N(AC)(BO-(A)(C)(BC)-(B)(C')(AC) + (A)(B)(C) (C)[i\r-(C')] 134 Oll the Theorii of Association Write {AB\=^{A){B), (4C), = -i(4)(C), (ßC')„=^,(ß)(0), subtract {AB)a froiii lioth sides of the above eqiuition, simplify, and we have Mm (An\ N[{AC)-{ AGU[{ BG)-{BC),-\ (AB) - {AB\ : ü [F -TQ] • 'I'hat is to say, tkere will he apparent assodation hetween A and B in the universe at large unless either Ä or B is independeiit of C. Tlius, in the iniaginary case of inhcritance given above, if A and B stand for the presence of the «attribute in the parents and the offspring respectively, and C for tlie male sex, we find a positive association between A and B in the universe at large (the poolcd results) becausc A and B are both po.sitively associated with C, i.e. the niales of both generations possess the attributo niore fre(iuently thaii the females. The " parents with attribute " are mostly males ; as we have only noted offspring of the sanie sex as the parents, their offspring invist hc mostly niales in the sanio proportion, and therefore more liable to the attribute than the niostly-fomale offspring of " parents without attribute." It follows obviously that if we had found no inheritance to exist in any one of the four possible liues of descent (niale-male, male-femaie, female-male, and female-female), no fictitious inheritance could have been introduced by the pooling of the four records. The pooling of the two records for the crossed-sex lines would give rise to a fictitious negative inherit- ance — disinhoritanco — cancelling the positive inheritance created by the pooling of the records for the same-sex lines. I leave it to the reader to verify these Statements by following out the arithmetical example just given should he so desire. The fallacy might lead to seriously niisleading results in several cases where mixtures of the two sexes occur. Suppose for instance experiments were being made with sonie new antitoxin on patients of both sexes. Tliere would nearly always be a difference between the case-rates of mortality for the two. If the female cases terminated fatally with the greater frequency and the antitoxin were administered most often to the niales, & fictitious association between " antitoxin" and "eure" would be created at oncc. The general expression ioT{AB) — {AB\ shews how it may be avoided ; it is only neces.sary to administer the antitoxin to the same proportion of patients of both sexes. This should be kept constantly in mind ;us an essential rule in such experiments if it is desired to niake the most use of the results. The fictitious association caused by mixing records finds its counterpart in the spuriiius correlation to which the .saine proeess may give rise in the ca-se of continuous variables, a case to which attention was drawn and which was fully discussed by Professor Pearsoii in a recent momoir*. If two separate records, for each of which the correlation is zero, be pooled together, a spnrious correlation will necessiirily be created unless the niean of one of the variables, at leiist, be the same in tlie two cases. • Phil. Tram. A, Vol. 192, p. 277. A FUßTHEK STUDY OF STATISTICS RELATING TO VACCINATION AND SMALLPOX. By w. e. macdonell, ll.d. (1) The followiiig papor is a continiiiition of the Note publishcd in Biometrika, Vol. I. No. 3, but I have beeu able to extend my lesults to a comparison of the differential chaiacter of tvvo epidemics in the sanie city, and to some preliminary cousideration of the intiuence of oceupatiou on the existence or non-existence of vaccination and on the severity of a smallpox attack. My Statistical raaterial is taken from : A Summary of Statistics relating to Vaccination and Smallpox as observed in the Cases admitted to the City of Glasgow Smallpox Hospital, Belvidere, hetween lOth April, 1900, and mth June, 1001, by Dr R. S. Thomson and Dr Fullartou — a paper read before the Royal Philosophical Society of Glasgow oa 2ud April, 1902 — and from supplementary statistics relating to the same epidemic kindly sent to me by Dr Brownlee, Physician Superintendent of the Belvidere Hospital. The tables were prejDared, and the coefhcients of correlation with their probable errors calculated in the same manner as in the earlier paper. I may repeat that I use " mild " as equal to " discrete," and " severe " as equal to " confluent " and " haemorrhagic." The few cases (only 2i per ceut.) of " doiibtful " vaccination have been excluded from my tables. The following are my tables for the Glasgow epidemic of 1900-1. TABLE I. Recoveries Deaths Totais Vaccinatoil ... Unvacciuati'd 149:5 5ü 150 63 1643 122 Total« 1552 213 1765 ;•= -6294 ±-0296. 130 Vaccination and Smallpox TABLE II. Mild Severe Tnbds Vaccinated... Unvaccinateti 13C1 53 282 69 1643 122 TotaLs 1414 1 351 1765 r= -5 162 + -0322. TABLE III. ' Sears Mild Severe Totais Fovea ted ... Unfoveated... 573 788 82 200 655 988 Totais 1361 282 1643 r= 1929 ±-0374. TABLE IV. Area of Scar Mild 1 Severe 1 Totais 1 Over half Square inch Half Square iueh and under 805 556 113 169 918 725 Totais 1361 282 1643 r= -2646 ±-0301. TABLE V. Xuinlier of Sears Mild Severe Totais Twü and upwards One 652 105 709 177 757 886 Totais i;561 2S2 1643 »• = ■1511 ±-0306. TABLE VI. Vaccinated Cases. Recoveriea Deaths Totais Mild Severe 1336 25 157 125 1.361 2S2 TüUls 1493 1 150 1643 r = -8603 ±-01 50. W. K. Macdonkli. 137 'l'he stalisdcs also ciiablc nie to lind Llic cüetKcieut of correlatiou betwtjcn agc am! severif-y of attack. TAÜLK Vir. VacciiKited Cases. Years Mil.l Sevei'e 'l'otal.s (1 ti) .'() .'" .111.1 ovcr... l'I 1 1117 11 26s 258 1385 Totais 13G1 282 1643 »•=•3693 + -0386. If tho divisioii be made at 25 years instead of 20, r = -3297 + -0314 ; if at 35 yeara, r = •3180 + •0293. Caiculating this cocffieieiit for the Glasgow epidemic of 1892-05, I find ?• = •33Gß + •0715, for a division at 20 yoars ; I am nnablo to calculate it for a division at 25 or 35 y(,'ars, as thore an? iio divisions at these periods in the statistics of the earlier epidemic. (2) For the sake of comparisoii, the coefficieiits for the two Glasgow epidemica are collected in the table below. TABLE VIII. Coedicient of conclatiou betweeu Epidpiiiic of l'JOO-l Epidemie of 1892-5 Vaccination and .strength nf rosi.staiu'e Vaccination and severity uf attack Fove.atiün of .scar and severity of attac-k Area ofscar and .severity of attack Number (if «cars and severity of attack Strength of re.si.stance and severity of attack ... Age (division at 20 year.s) and .severity of attack Age( „ 25 „ ) Age( „ 35 „ ) ■6294 + -0296 •5162 + ^0322 •1929+0374 ■2646 + •OSUl •1511 + -0306 •8603 + ■Ol 50 ■369:5 + ■OSsij •3297 + '0314 •3180 + ^0293 •7783 + ^0365 •9123 + ^0181 •:5951 +^0594 •352(1 + ^0584 •2323 ±-0616 ■3366 ±071 5 It will be observed that in the recent epidemic the coefficient of correlation between vaccination and strength of resistaiice, while less than in 1892-95, was very much the same as in previous epidemics in other towns*. On the other hand, the correlation between vaccination and degree of severity of attack is very much less than in 1892-95, which points to a marked difFerence in the character of the two epidemics, the earlier being the milder. Also the correlation between degree of severity and (1) foveation, (2) area, and (3) nuniber, of scars is less even than in 1892-95; but I understand from Dr Brownlee that in conclnding whether a scar is good or bad, clinically, he would take into consideration its area. * Biometrika, Vol. i. No. 3, p. 380. Bioriieti'ika it 18 138 Vaccinatiou and Sinallpox the area of foveatioii, and the amoiint of depression and " puckering." The fact is also clearly broiight out that adults are considorably uiore liable to the severe forms of the disease than the young. (3) The next part of niy intjuiry preseuts some features which, I think, are novel, and is based on figures relating to the Glasgow epidemic of 1900-1, supplied by Dr Brownloe. He grouped the male patients according to their occnpations, adopting the grouping of the Registrar-Geiieral's Reports with ihree modifications : (1) Pi'ofessions, Clerks, etc. are all grouped together, as the number of patients other than clerks was very small ; e.g. only four professional inen were adniitted to Belvidere : (2) Railway nien are shown separately from Transport Service : (3) Shopkeepers include all kinds, instead of the limited number adopted in the Registrar-Geueral's Reports. Cases of doubtful vaccination (25) are again excluded. The following table is prepared from Dr Brownlfo's figures : TABLE IX. Smallpox Cases admitted into Belvidere Hospital 1900-1. (Males.) Vaccinated Occupations Unvaccinated Recoveries Dcftths Totais llinei's 25 3 28 2 Lalumivr.s 167 25 192 9 Metal Workei'« 1.59 14 173 Shoiikeepei-s 70 10 80 5 Kailwav inen 13 2 15 1 Tninspnrt Service ... 85 11 96 3 Otlier Trades 4ß 10 56 3 Si)irit Siilestnen U 3 14 l'rofessioiis, Clerks, etc. G5 C 71 l Building Trades 74 9 83 1 Textiles 22 1 23 Unoccupied 10 1 11 — 842 1 29 Totais 747 95 Froni the statistics of Table IX. I wanted to svscertain the correlation between the social status of the patients and (1) the presence er absence of the scar, and (2) in the case of the vaccinated their power to resist the disease. For this purpose it was necessary to divide my material into two chisses having a higher and a lower Status. No donbt considerable diversity of opinion may exist as to the eoniponcnts which should fall into these two classes. I consultcd from this Standpoint the death-rates of various classes as given by the Registrar-General. W. I\. Macdonki.1, 13!) My chicf tlitticiilty amsu iVoin thc group of nicliil workers who form by f'ar Lhc largcst siiiglc occupation group after fche labourers, and yet exhibit not a singlo unvaccinated person. It is clcar that such a class according as it i.s included in t.hc higher or lower Status group niay coinplctely changc the correlation botwoen Status and vaccination. If tho metal workei's inchided filecutters and Icad- workers, thcy wnuld have a high dcath-rate and wc (»Light tu put theni in the group of lower status. On the other hand in sonie districts the metal workers bclong to the most trained and the best paid elass of craftsnien, men who are likely to bc well iii)urish(Ml and with fairly heaithy luinies. I accordiugly wrote to Dr Brownlee iii(|uiring as to the status of the Ulasgow metal workers and as to thc abseuce of unvaccinated eases among theni. His reply is as foUows : " No fileeutters or leadwurkers were included in the metal workers. The lalter "were in great majority made up of shipbuilding and forge employees, these " being the main metal Industries in Glasgow. A very few brassworkers were " included. The status of these metal workers as we received them was not so high " as that of either the men who work on the Railways or in the Building Trades, " and I think that they should be included in your second class. The abseuce of " unvaccinated cases is to a certain extent explainable. Most of these workers were " from the neighbouring forges where on account of thc contiguity of the hospital " the employers have for a long time exercised a certain amount of supervision of " the vaccination of those employed in the works. Smallpox has several times " invaded these works since 189Ü, so that I think unvaccinated persous must " be very few indeed." It must be at once confessed that this rnore or less enforced vaccination of the large group of metal workers is a very disturbing factor in the consideration of the relatiou between status and vaccination. Adopting Dr Brownlee's view I first put the metal workers in the class of lower status. As a tirst grouping I took for those of higher status : Professions, Clerks, etc., and Shopkeepers. I obtaiued : TABLE X. First Grouping. Class Vacciiiated Unvaccinated Totats Lower Higher 691 151 20 9 711 160 Tntais Sl:^ 29 871 •1862 + -0737. I now included in the class of higher status, Miuers, Railway men and Building Trades as well as Professious, Clerks and Shopkeepers. 18—2 14U Vaccination ainl Sinallpo.' TABLE XI. Second Groiipiug. Claas Vaccinated Unvaccinated Totais Li>\ver lliglicr L'77 13 .•,si Totais 842 29 871 1 (• = •1362 ±-0695. 'I'hcre is tlius for both cases a quite sensible if not very large correlation betwoen Status and vaccination,— CK.ses of unvaccinated persans occur more frequently in the classes of higher than of loiuer Status. Putting on onc side Dr Brownlee's views on the Status of the nietal workers I now formcd two tables in which the nietal workers were included in the group of higher Status, adding theni first to Professions, Clerks, etc. and Shopkeepers, as a third grouping, and tinally to all tliese togethcr with Miners, R;iil\vay men and Building Trades as a fourth grouping. I found : TABLE XIL Third Grouping. f'la.ss Vac-c'iijatC'l ITnvaci-iiiated Totais Lower Higher 518 324 20 9 538 333 1 Total:^ 842 29 871 ,•=-•0872+0713. TABLE XIIl, Fourth Grouping. ClilS-S Vaccinated Unvaccinated Totais Lowi-r llighor 3!I2 ' in AM 1.-. 108 m;3 Total» SIL' 29 871 »•=-•0939+0691. The correlation has clearly swiing round, and therc is uow a very slight correlation, hardly sensible considcring the probable error, between higher status and vaccination. In view, howevcr, of ])r Brownlee's opinion as to the cliaracter W. R. Macdon Ki.L 141 of thc mctal workcis, I do uol thiiik Ulis Classification is legiiiiiiate. Oonsidcring also tliat vaccination appoars to bc morc or less compiilsüry aiiioiijf a coiisiderablc scctiiiii ol' Uiciii 1 (K'lrnuiiR'd 1:1) (iiniL tlicni all(jgcl lirr, aml thca obtaiiied tho i'dlldwiiiL;' table: TAIU.K XIV. First Grouping, MetaL Wi)rkers exduded. (.'lass Vaccincated Unvaocinatod Total« Lowor lli;4her :, 1 s I.M 21) !) :.:!S Kii) Totiils tiuy -2^) (i;)8 /■= 1107 ±-0703. Compariiitf this with Tables X. and XII. we see that there is a sensible bat siiiall correlatiDU betwecii bitrher status and iinvaccinated coiidition. Now I lay no partiuular strcss un tbeso results beuausc the nialcrial is t'ai' tot) sparse*, bat I believe that the abovc statistics are tho only ones hitherto dealt with with a view to dctcrmining whether the classes of bighcr status — presumably the better fed and healthier classes of the Community — are or are not more frequently vaccinated than the lower, presumably the less nourished and less healthy classes. No dogmatic couclusion can be drawn from these data, but they exhibit no evidence at all for the unvaccinatcd class being of lower status than the vaccinated class ; on the contrary, tliere is slight evidence to show that the unvaccinated in Glasgow occur rather more frequently in thc classes of higher Status. (4) I turn now to the question whether among the vaccinated there e.xists a correlation between status and severity of the disease. I obtained the following tables : TABLE XV. First Grouping. Class Deaths Rocoverie.s TotaLs Lower Itigliei- 79 1() fil2 i;ir, • (i91 ir.i Totais 95 747 842 !• = -0249 ±-0566. * The fewnes.s uf unvaccinated cascs iiossibly arises fnim tho fact that diiring thc 1802-95 cpidemic in Glasgow vaccination was ])erfiinned on a large scale amongst all classes, so that the cpidemic of 1900-1 fouiid thc t'reat majoiity of the population vaccinated. 142 VaccinatiuH atid Smullpox TABLE XVI Second Groupiiiij. •'••- Dcalhs Kci-ovcrie.s 'l'otals Lowcr Higher ... 65 500 30 217 565 277 • 'l'otal.s ■■<■• 'i IT ^,. , /•= -0216 ±-0500. Th;it is to say there is a positive bat practically insensible corrclation between Status and power to resist the disease. I now included the iiietal worker.s in the liifrlitT statns and fuund with the former groiipings : TABLE XVII. Thinl firuiijjiiiij. Class Deaths Recoveries Totais Lnwor Higher i;.-. i:.:', Totiils 95 1 747 842 . /•= 1086 ±-0490. TABLE XVIIL Fourtk Grouping. n,-,.s^ Peaths ' Ri'ci.vcries Totais Liiwcr Higher 51 44 341 406 392 450 Total.-, !)5 1 717 1 '-lä r = -1052 ±-0478. Thus the con-elation between better statns and recovery is slightly increased. Leaving the inetal workers as before out of consideratiou, I find : TABLE XIX. First Grouping without Metal Workm'S. Class h.;itli. Recoveries Totjvls ' Lowcr Higher 65 16 453 1 3.^ 518 151 T..t.,l< '^I 588 609 ,— ■U579+U 594. W. U. Macdonkll 14:5 This is probably the iiiost satisf'actory result as abüut the mean of tlie prcvious tables. We accordiiigly concliule that thore is a very slight relation betweea Status and recovery froni attaek. The way in which the inetal wcji-kcrs iuerease tbis correlation wlien they aro ])lai'(Hl in the better class is reniarkable ; they have in Glasgow a eoinparativeiy .sniall niortality trom sniallpox, yet we und f'roin tho ')5th Annual Report für Enylaml of the Registrar-Generul — Medicul Siij)j}leineiit that the general mortality among nietal workers is higher at all ages, espceially alter So, than that of occnpied inales. 'I'he tigures are as follows, if the Standard rate of mortality among oceupicd inales at, each age be taken ns 100: Ages 15— 20— 25— 35- 45— 55— 65 and up. Metal Workers 105 lOG 103 111 122 129 12.S It would be interesting to conipare the special classes gronped as metal workers in Glasgow, with those enibraeed undcr the sarne heading in the Registrar- General's Reports. (5) Conclnsions. (i) The Statistical constants for vaccination and sniallpox difter sensibly for the same place with two different epidemics, i.e. epidemics seem to be differentiated in character. (ii) The statistics of Glasgow do not indicate that those of lower statns — and therefore probably worse nourished and housed — provide the bulk of non- vaccinated cases. On the contrary there seems to be a slight tendency for the non-vaccinated to be of higher statns. (iii) There is a slight althongh scareely sensible correlation between statns and power to resist a sniallpox attaek. The Glasgow .statistics do not go very ftir, but as far as they go they do not justify the Statement: That the apparent protection of vaccination is dne to the nnvaccinated belonging to classes of lower statns which have a far smaller power of resistance to the di.sease than the better nourished classes of a higher statns in which the members are more generally vaccinated. I have not dealt with the statistics as to female patients as the great bulk of them are classified merely as " Housewives," which throws no light on their social Position. In concludiüg this paper I venture to express the liope that the statistics uf the recent London epidemic may soon be issued and in a form which will adniit of due consideration of the problems referred to in this paper. Their magnitude gives them extreme valne and their publication is no doubt anxiously awaited by a wide circle of scientific iuquirers at home and abroad. 144 Vacciiialion and SuuiUpox NOTE. Froiii thc "Times" of 20 November 1!JÜ2, il iippeurs Llial llie iVledical Urticur of Health for Islington has prepared his tiual report oii the 1901-2 epidemic so far as his own district was conccrned, but I havc becii unable to obtain a copy of it. Tlie paragniiib in tlio " 'l'imes ", however, ciiables mc to form the foliowing table : TABLE XX. Smallpox in Idingtuii, 27 August 1901 — 29 August 1902. IvProvorios | Dp.-iths Totals Vacciiiiitcd Uuviiccinated... 207 35 29 30 2:j(j 65 Totals •242 .09 :',ul ,•=•5744 + -0560. This vahie of ?• is in extrenuly close agreemcnt with that obtained for 1017 cases which were " completed " in London in l!)ill ♦, viz. r = ".^779 + 'Ü-'U!. To show that tho eloseness of the agreenient h(jlds tliroiighout the investigation, I give the eqnations from which r \va.s calculated in both cases: For the abovo .SOI cases : •07OG08 /•«+ 002326 r +\r,l-^U) ,■* + 017030 7-^+ -336384 ?•» + r = -707743. Für the 1017 cases : ■071G07 i" + 001780 c" + •14963Ü /-^ + 0198442 >•» + 326092 r- + r= •710100. * Uiomrtriha, Vol. i. No. i. p. HT'.I. C()01>ERATIVE INVESTIGATIONS ON PLANTS. IL VARIATION AND CORRELATION IN LESSER CELANDINE FROM DIVERS LOCALITIES. (1) In view of the dat;i t'or F. nniancaloides published by Dr F. Ludwig* and Prof. MacLeodf and the Statistical coiistants detenuiiied für theni| it seemed desirablc to obtaiu rather iiiore Statistical inaterial and a more comprehensive series of constauts for the purposes of a coniparative study. There wcre two points to be cousiderod, naniely : (i) the infiuence of locality aud (ii) the influence of the tinie during the flowering season at which the Üowers were gathered. Unfortunately we wcre in the present season in no case able to obtain from our collectors two series from the same locality with a nionth's interval between the gatherings. All the collecting except in one case had to be done during the brief Easter vacation of our workers, and this did not adniit of a double gathering in the same locality at a suitable interval. The one exceptiou is that of the Bordighera collection. Mr Francis Galton most kindly offered, as he was wintering on the Mediterraucan, to provide a double series of Lesser Celandine flowers. The first series was gathered about Fcbruary 19th aud at once dispatched to England. On arrival it was fouud that practically every sepal, overy petal, and nearly overy stamcn had fallen Irom the flowcr-heads. This lost us our earlier series, but we learnt a most valuable lesson, namely : that transit of any kind, even by hand, will cost the flower if nearly fuU-blown one or more sepals, petals or stamens. Our plans had thercfore to be chauged ; each celandine flower was now gathered as a bud and wrajiped up in a small piece of tissue papei". This involved a great increase in the labour of gathering and a much greater one in that of counting, a good deal of which had to be done under a leus, but we were thus certain of preserving all the parts of the flower intact. Mr Francis Galton suggested and carricd out this arrangement in a second series gathered between March 4th and 7th which reached England safely, but three or four weeks later than this there were no flowering celandines to be obtained in Bordighera. Mr Galton's plan was carried out in the further collections inade in Guernsey, Dorset and Sun-ey, the collectors gathering the buds, and wrapping them up * Biometrikü, Vul. I. pp. 11—20. + Ihiil., pp. 1-2.5-128. J Ibid., pp. 31fi— 31<). Biumctnka ii 19 14(J Cooperative Inrcstifjatiotis on Plantn in situ, in a small picce of tissuc papcr. The buds dricd and wcrc prcscrved for wceks, to bc countod at Icisure. Even if a petal ur scpal feil oflf it was preservcd in thc papcr wrapper. \Ve believe it is largoly due to this nicthod of gathering that our results show such a totally dift'eient distribution of scpals and petals to thoso of other observers. Take thc casc of scpals. In thc Gucinscy scrics \vc found ouly thrce in- dividuals with Icss than thrce scpals. On cxauiination linder the niicroscope in one of thesc cases an abortive scpal was found, in another the sepal had clearly oncc been attachcd, but in thc third Civse thc hcad was unfortunately lost before microscopic cxauiination. Wc think Ihat wc niay .safcly affirni that no true case of less than thrce scpals was fouml in tiic Guernsey plants. In the Dorset- shirc gathering no cases of less thaii thrce scpals occurrcd. In thc Surrcy gathering thcre werc in the matcrial six cases of less tiian thrce scpals, and in five out of these six cases thc rulc of gathering bmis duly had been disregarded by a young collector* ; and thc sixth was an abortcd Hower in which thc staniens wcrc not propcrly developcd and thcre werc no pistils at all ! In the Bordighcra celandiues tiicre arc only two cases of less than three scpals. These have not been excluded from the calculations, becausc we had not whcn counting them leamt from thc Doreet and Gucrnsey scries to regard all cases of less than thrce sepals with grave su.spicion and cxaniinc such cases undcr the microscope. But if in 624 cases only two such individuais occurrcd, it seems cxtremely probable thivt evcn the.sc were cases of scpals knockcd off or abortcd. In a total of 21 -iO hcads, thcre were 11 cases of less than thruc scpals, two of these wcrc abortcd ilowcrs, fivc of them wcrc old Huwcrs witii parts loosc, one had ouce had a sepal which had been lost, and thrce were not closely examincd. The experiencc we have had leads US to believe that cach flowcr ought to be gathercd as a bud and at once wrapped up. It seems to us that all we cau admit is a possibiliti/ of three heads with less than three sepals in 214!) cases, while Dr Ludwig's 3000 from Greiz show no less than 60 defiuitc casesf. We feel fairly confidcnt that had the Greiz flowers been collccted as buds and possibly gathercd and counted by adults instead of school children they would not have differed so widely from our material in this respect. As far as our experiencc rcachcs, wc ((uestion the existence of any normal flower with less than thrce sepals. This vicw may be modified whcn fiutiii'i- matcrial from central Europc gathcnil in bud and, it' necessary, microscopically examincd, is availablc. Wc hope next se;vson that this problem of the sepal may bc directly iuvcstigatcd. (2) Material of the present iiivesti M 1-1 -4060 -8434 4-2562 4-8508 CO 00 .-1 Ol in 00 ^ tr- -1981 ■0815 -1150 -2924 in s CO cb coNxu5c»'t~'neocc-<*j CO •fl 'S i CO IS 00 eoooo^' ojcoeoo g +1^ +1 " +1" +1 1 CO ■-IC<1 ■ 1 1 1 1 1 1 , 1 11 +1 IS a a c 1 o Q t i b» -r ^ It •"• ^ ** ^' o pcoc:0— 0— in— oo-/:osi-.-iira O^^OOr^OO■MCO« OStMOlCiCO— pC2 rHOcD— coi-corj oicm-o-MOr; " +1" +l?3 +1^ +1 j ' +1 +1'" +1=° +1 rHO 1- OS ^1 (N ^? 71 CO -?! in ri CO — c^rpcrT^pc^c pcpc +1 +1 +1 +1 1 +1 +1 e O 1 Oh •-H o CO i *>- c. -«1 o IM CO c: c^ i^ 00 ri in t - CO c 00 i^ 00 ?? 1 ' os^i(N^:in— f-ii' t-'Mco^t^^t^'M occo — in-Ti-i/: eI>rco-05-■#^^ " •i-i'* +i?5 +\^ +\\ +1 •+!■* +i"° +1 •«" CO ■* (N c^ — 1 in CO lÖ t^ cö rHC T 1 o ^ 1 £^ 1 1 t~ 1 00 1 { od 1 " 1 1 1 ^M 1 1 0^1 ^H iQ Tvl 1 c^l 1 X % 1 t^ r— 1 1 1 tN 1 CO ^ rH O O 1 •Tt* 1 o 2? 1 t^ Oi 1 1 1^ 1 CD 00 1 CO 1 CD s l§ M O CD ^ IS-" 1 O ^ 1 00 -f 1 1 « 1 1 CD M 1 11 1 r^ ^ 1 1 l-H 1 r— ' 00 ^ i !=^ M C35 1-1 MM "^t CO 1 CD (M 1 00 1- M ■^ 1 - M o •s^dag o lO >o o '■V' -* - 1 - S5- 1 1 1 ^ ^ ! -- - ^ :c 1 n § X [ X l§ o 1 CD So ^ CD :^ '— ' 1 O ^ CC [ CO s? 41 1 Ol Ol 05 ^i Ol n -t 1 ^1 1 Ol -^ « 1 CC C5 "0 ^'-" s J-. •>! 1 ni '< -f 1 h'. ^ il ^ 1 ^t n 1 X '^i 1 iri 55 jvj ^ 15 ci öl o 1 2 g :2 1' :=! nH 1— 1 oc CO 1 CO ?- T--) Ol 1 CT »-H I 1 1 '-0 1 1 j - 1 - -; — o •si'edgg 1 § S-: - - 1 1 1 .^ Ol 1 Ol [ oq So -f 1 -t ^ ■o 1 o CO fM 1 1 S ^1 ' -M y -f -ri [ CC n ^T S 1 1 1^ — 1 TT ' n 00 I5 n 1 1 n a 1 1 i S! 1 1 ^1 00 1 1 00 i 00 1 1 CD g -l^— ' t^ ;? ^ 1 1 iL l-H 1 1 r- \ I - - § (N 1 1 IM 0-, 1 =0 1 1 !0 1 «5 1 1 »5 " 1 1 r- C- «).l.- ■r. CO I— I (72 I— I ö j W m O X C X cc IN CD 1- ' - 1 M 1 1 i§ 1- 1 1 ^H S? MM 1 f- 1 -^- 5; l 0; 1« M BH 1- M r-l ^ MM 1 n 1- 1 1 »-* § l»' 1 1 (M s; MM 1 § MM ^ M 1 i - i§ l<^ M fN 1; i « 1 1 I- 'S MM W5 MM ?T !-^ 1 ! ■0 IM ^ 1?) - "" ~i '^J ! m ■■ 1 a \ ^ n\ 1 1 « 1 CO 1- a5 CR « 10 1 00 j u i IS'- 1 00 ~4 i:t^ 1 I- »> iS"' oc CO 1^ 1 t5 1« M 03 ;^ r; ■ ■ « c^ - " 1 1 " > MM '-- ' _ ■ ■ «i »2 ■<>■'.■; 1 X! •spdog X 1 - , - 1 1 1 ! 1 % 1 1 1 % 1 1 1 % 1 1 1 a\ \ (N 1 1 1 ^ KI- "* ^ 1 1 1 ^ (N 1 04 1 1 1 ^ (N 1 (M ^ -r 1 't ^ CD 1 CO 5^. 1 ^ 1 10 ^^ 1 l§ "*• 1 >o ^ 1 " 1 Oi r«. m; «5 ec 05 CD 1 CD ^ 't;; g 1 g 00 1 00 g fM IN CJ 1 i ^ -- CC CO II « 1 ec 1 5; g- S i oc 1 (N 1 g 00 1 (N 1 00 00 *-< !M 1 n \ 1- 1 CO S — CO t~ 1 t— '-1 1 s -r [ gl ..•5 i ■: -r c§ S § 2 ■9[Bdag •s[tjdog II. Variation ancl Correlation in Lesser CelamJine 157 8 ^ ^ 1 r-H ^ - 1 1— 1 (D 1 CN ^ 1 ^ - 1 l—t 1 1 1 tC I •* C5 •* .-1 lO 00 SO « 1 CO so M r-< 'J' ^ CD 1 CD ^ ^ 1 -^ ^ n 1? o 1 lO n 1 1— 1 1 c^ so CO —1 05 i o 1 T— « (N ■-< CO ^ t^ [ I- 0-! CO ^ 1 1—* ' -+ ^^ 2 1 1 -1* 1? 2 1 1 CO '^ 1 1 1~ O ,-1 1 CD CD CM CO 1 1 ■^. CO ^ 1 CO 1 CO C3 2 1 1 a> ?, 1 1 00 CM CO 1 1 CO 1 1 CO CO § CD ^ 1 Q-1 1 fM § C-1 1 C2 ^ g 00 ^ 1 tN 1 CM 1^ 00 rH 1 CM S3 S 1 1 OE 15 2 1 1 ^ 1 1 kO 1^ CO t- 1 1 t^ 00 1 1 00 •o 1 1 o CM 1 1 CM ^ 1 1 r—' 50 »-1 CM 1 1 CM ©? -^ 1 1 -H -- 1 1 1—1 1 O 1 1 1 Ci -- 1 1 - "0 *^'-'j 7} 2 W >-H ^ < ö H «J Yl .«i [^1 1^ s 0) 3 < «5 CO •Jl ^ h-l 1 — ( t-H W I^H r3 l — ' 05 CM O f-' --' ^ CD ^ O CD 1 1 i i - l - 1 1 1 ■"* 1 T 1 1 ^ • o: >^ (NO Oi <>• Pf •-. ©i 1-2 M 1 1 M 1 1 1 1— ' r § MMN 1 Ol V 1^ 1 1 : I ^ ^2 f- =C •>. 1 CA q a S hH S > m X •SIB19J 3. j2 Tt CO t^ M CT ■* CT 00 — CO ro CT ^- i ::; 1 1 1 1 1 1 ^ T § 1 1 1 1 1 1 1 § 1 1 1 1 - 1 1 - § 1 1 1 1 1 1 1 U3 1 1 1 1 1 1 1 § 1 1 1 1 1 1 1 !§ 1 1 1 1 1 1 1 ^ s? 1 1 1 1 1 1 1 1 1 1 1 1 1 1 •5 1 1 -- I 1 1 1 fH 1 1 1 CT 1 1 1 CT § 1 1 1 —1 CT 1 1 CO Ol 1 1 CO 1 1 1 1 CO 00 1 « CT .- eo 1 1 03 1 1 - 1 - 1 1 CT ^ 1 1 r-CTr-i 1 1 ■* 1 CT « -H 1 1 1 CD 1 rt CT CT CT 1 r- 00 ^ 1 00 -* ■* CT 1 1 00 , •* t~ re CT 1 1 CD 00 t— t 1 00 eo -t CT — 1 1 eo ■* CT 1 1 03 CT 1 03 C^ 1 1 1 CT CT 1 C3 -* CC « rt 1 % *- 1 t^ 00 P5 rt 1 1 § 1 T)< CO -* CT 1 1 CD CT CD CO CD CO IO.-CO« 1 , 1 CT t~ CT 1 I 'CT II 1? 1-1 IS >0 10 CT 1 1 00 CT C^ t^ CT 1 -1 1 s 1 CO 1 i-< f-t 1 CT ^ . t^ »— 1 1 1 CO CT i^ r-1 — -* 1— 1 1 1 t~ ?5 |CT^^ 1 1 1 f— i ?( 1 CD^^ 1 1 1 00 ^ 1 COi-H ^ 1 1 1 00 ^ 1 10 CT 1 1 1 1 t~ ^ 1 lO 1 1 1 1 1 ir 1 'S 1 1 1 1 1 1 ^ 1 1 I 1 1 1-^ 1 - 1 1 1 1 1 rH § 1 X Pm m p^ 1 r-. 05 (M p r-. ,-. ^ M -^ 1 1 1 - 1 1 7l r— 1 MINI i 1 1- 1 M n 1 1 1 1 1 1 1 1 1 ■ 1 n M'j^ 1 1 1 n ^ 1 1- 1 M 1— t 1 M ! 1 1 1 ^ 1 1 1- 1 1 >-i n 1 1- 1- 1 CM SO 1 1 1 1 M 1 § 1 M- 1 1 - 15 II 1- 1 1 - ^ 1— 1 f— 1 -i iy\ ^ CO \ 1 ^ 1 1 CO 00 1 « CD 05 1 1-H Ol 1 1 t^ t-- l-H 1 1 1 r^ 1 03 1 Ol CO CD -— ' 1 1 i> 1 Ol 00 in • r-^ i-^ r^ 1 1 1 CD II 1^ 1 ^ CO (M CT 1 ! CD ! 00 CD 1 1 CO rH I 1 1 1 i Cb >^ 0-1 sjB^aj t- 1 1 1 1 1 1 1 1 Co 1 1 1 1 1 1 1 1 1 1 fM 1 -« 1 1 1 1 1 1 1 1 1 ••SO 1 1 OJ rH rt 1' T ^ 1 1 M 1 1- 1- 1 ^ .11111 1 ?^ l<^ 1 1 1 1 1 1 (M 1 1 ; u n 1^-^ r CO ^. 1 so 12 M 1 o rH l-' M 1 rf 1 -^ II 1 lO !^ c>o c~. ::i ^ o H •sitr}3 i 1 .-1 oq 1 1 «3 1 IS 1 M >n •^ »^ 1 1«- 1 1 O «5 1 1"« M 1 lO M" 1 M n I-« 1 1 1 ■* I 1 =^ I M (N ■ ■ • ' - i :: o H CO O >< X •81B?3J -r c<3 t~ e^ (M TT — ( 91-8S 1 1 1 1 1 1 1 1 ».-3 1 1 1 1 - 1 1 I-H 99-^9 1 1 1 1 1 1 1 1 W5 1 1 1 1 ^ 1 1 — W5 1 1 1 1 1 1 1 1 ^■3 1 1 ^ 1 1 1 1 I-* § 1 1 1 1 i 1 1 1 ^ 1 1 1 1 1 1 1 1 :§- 1 1 ^ 1 1 1 1 p-< 1 ^ 1 1 1 1 1 r-t ^ 1 ^ 1 ^ (N 1 1 rf ^ 1 »H 1 (N 1 1 1 CO :? \ ^r- \ 1 1 1 o ^ 1 eo -- ^ 1 1 1 X 1^ 1 ?? 1 CO (S 1 ■* 1 1 a 03 1 CO e^ 1 (N 1 1 X ^ 1 CO r- ■* 1 1 1 I—" ^ 1 o CS o -. e; »3 »"i »*H »^ **H 1 •S|B)8J II. Variation and Corre/ation in Lesser Celandine 1(1 O Ph 1 CT CO ~ . : : ! 1 1 1 I 1 1 1 1 1 1 1 1 1 j i 1 1 1 -- j 1 1 1 1 1 1 1 1 1" M 1 1 1 1 1 1 M 1 1 1 1 1 1 1 1 1 1 i > 1 1 1 M 'S:. 1 M 1 1 M 1 1 M^ 1 M 1 M 1 M 1 1 M 1 M 1 M M ^ 1 M M 1 M 1 1 M M 1 1 M 1 M 1 1 1 1 M 1 M 1 M 1 IIIII-III-MMIMIIMIIIIMIMM- n 1 1 1 1 M M -^ 1 ^ 1 1 1 1 ! 1 1 1 1 1 1 1 1 1 1 1 M M 1 1 CT 1 M M 1 1 -^ 1 1 1 1 1 M 1 1 M 1 M 1 1 1 1 1 1 1 1 1 M ,- ^^ 1 1 1 1 M 1 1 1 1 1 1 1! 1 1 M M 1 M 1 1 1 1 1 1 1 1 1 1 1 ^ 1 1 1 1 M -- 1 M 1 M 1 ! 1 M 1 M M M M ! M M 1 - % 1 1 1 M M M 1 ! 1 1 1 M^ 1 1 1 1 1 1 1 M 1 1 1 1 1- 1 n^ ^ 1 1 1 1 1 M M 1 1 M M 1 1 1 1 M 1 1 1 1 1 1 1 1 M 1 1 % i 1 M 1 1 1 M M 1 1 M i 1 1 1 1 1 ^ M 1 1 M 1 1 M 1 -^ ^i 1 1 1 1 -^ ! I 1 1 1 1 1 1 1 M M 1 M 1 M 1 1 M 1 1 1 1 1 I— 1 CT o1 1 1 M 1 1 1- 1 1 1 1 1 1 1 1 1 1 1 1 1 1- M 1 1 M 1 1 i 1 ?- 1 1 1 1 1 1 1 -^ 1 1 1 1 1 j 1 i .-1 1 IT) 1 1 1 1 1 1 1 1 — 1 1 1 1 1 ;^ M M 1 1 1 1 1 i" 1 1 ! 1 1 1 1 M i 1 1 1 1 1 M 1 1 1 1 1 -1 'S! 0) 1 M M 1 M 1 1 I-- 1 1 1----- 1 M i 1 1 1 1 1 1 1 1 CO M 1 1 i 1 i 1 - 1 1 Ol - 1 - i 1 1 1 1 1 - 1 1 1 1 1 ! 1 1 1 ! i 1 \ '-^ \ 1 l'^l |'-<-H|iNoq|.-i^.-iM|n!-1'tM|orH(M,-(|,-l| 1 1 1 1 1 1 1 1 1 1 1 m M 1 1 1 l^^l'-'^WOr-IOOS^O'^iOi-'l 1 1 j 1 1 1 [ [ 1 [ 1 1 1 1 CD 1 1 i-l 1 1 1-^ IM O O 05 1.-5 r^ -l< T< CD CT « 1 | '^ 1 | 1 | | 1 1 1 | 1 | 1 1 1^ 1 l^j_i^^cCl~«.^035 1 1 i-i 1 1 0-1 ^ . . 1 1 1 i 1 1 M t 1 1 1 1 1 1 M 1 1 1 M 1 1 1 . ! 1 1 1 1 1 1 1 1 1 1 M 1 1 M 1 M M 1 1 M 1 1 M 1 1 1 1 1 M 1 1 ! 1 1 M 1 1 1 1 1 1 1 M 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i M 1! 1 1 1 1 1 »> 1 1 1 1 1 1 1 1 1 (M 1 1 M 1 1 i 1 1 1 1 1 1 M 1 1 1 1 1 1 1 1 M 1 1 1 1 1 1 IIIMIIIIIIIMMI'-Illl«^llli-Il •>* 1 1 M 1 1 1 1 1 1 M 1 M 1 M 1 1 1 1 1 1 1 1 M 1 1 1 1 1 M 1 1 1 1 1 1 1 M 1 1^ 1 1 1^ 1 1 1 1 M 1 1 1! (N M M 1 1 1 1 M 1 1 1 1 1 1 1 1 M 1 M 1 1 1 1 1 1 1 1 i 1 M 1 1 1 1 M 1 1 M 1 1 1 1 1- 1- 1 1 1 M I M (N "^ 1 1 1 1 1 1 1 1 1 M 1 1 1 1 1 1 - 1 - 1 - 1 -- 1 M 1 1 1 -* 1 1 1 1 M 1 1 1 1 1 1 1 1 - 1 1 <« 1 1 - 1 1 1 - 1 - 1 1 1 CO 5^ 1 1 1 1 1 1 1 1 1 1 1 1 <« 1 1 M -- 1 1 1 - 1 - 1 1 1 1 1 1 o M 1 1 1 1 1 1 M 1 1 - M 1 - M 1 - - 1 1 1 1 - 1 M o ^ 1 1 1 ! M ! ' M 1 M 1 1 1- 1---- 1 1 1 1 1 1 1 1 in 1 1 1 1 1 1 1 1 1 1 |'^|<>^n| 1 1 |^Ri->r->(M|i-i| { 1 1 1» t— t 1? 1 1 1 1 1 |'~'l ! 1 |'-l|-*rt,-lrH| 1 1 1 1 OS ;: 1 1 1 1 1 1 1 1^1 |'-'-ip-| 1 1 TT 1 1 1 1 1 |rfrt(Mr-iT)5eo-Hinr-i«|oj|— 1— 1|>-'| 1 1 1 15 1 1 1 1 1 1 1 («N— ''-"'N'" — '•~|'^''-''-'0''N'-'| 1 1 1 1 1 1 ' IS 1 1 1 l""! 1'-' — -)'^i«'-''-'W"'-''^|'-'| 1 1 1 I 1 ' n ■^,• 1 1 1 1 1 |(Mr-(NT)'r-e>!ia-*-*|(N.-i(N'I1-IIIIIIIII-1II1MIIM m •-< "^ -1 -^ -^ ^ Oi Ol <>( 5) Ol Qi ei S( 0) '>( »3 »; 03 »i »3 »j 9^ «^ a; >: •^ -JJ- •-■T- -^ •suauiB^g Tl. Variation avd Correlatioti in Lexfter Celandine 163 ^ j^ 1— I 1^ ■Ji 1 II -— i—irMfMf.jrNOl-tffCCO-tCO'M'M '>■!'?)— ^ I l' -* ! 1 1 M 1 M 1 1 1 M 1 M M 1 M 1 ! 1 1 1 1 M 1 1 - .1 M " 1 1 1 M M 1 M 1 1 1 M M 1 M 1 1 1 1 1 M 1 M M 1 1- r 1 M M 1 1 1 1 1 1 1 1 1 1 M M 1 M 1 1 1 1 1 1 I-- 1 1 M M 1 1 1 1 1 M 1 M 1 M M 1 1 1 M 1 1 1 1 M M 1 1 1 1 1 1 1 1 M M 1 1 M 1 : 1 1 - MM 1— 1 M M M M M M M M M M M M M M 1 M 1 M M M M M M M M 1 - M M M --' M 1 M 1 1 - M §? 1 M M M M M M M M M-- M M M- M M M 1 n M M 1 M M M M M M ^ M M 1 1 0" M M M M M CO ^ M M M 1 M M M M 1 M M- I-- M M M M -* ^ M M M M M M M M 1- M-- 1- M 1- M M M 1 05 § MMMI!MMMMM»^MMMMMI-I-M Tf ^ 1 1 [ [ j 1 1 j 1 1 j 1 1 jn-*[r-l,— lr-lfNf-lr-(] | [ 1 l'-'j l'^O 1 j j - ^ ^ M M M M M- 1- M-- M M 1- M M M M M M o MMMMMI'~'"^I'-'!I''^'^I=^|^I'>'MM'^MMI « M M M M M M M M - M -^-^ 1 - M -- 1 - M M 1 Oi § 1 1 1 1 1 1 1 i 1 M^l l<^'l M'M'^'-''-h|co| MI'-'I I 1 1 1 1 o Si ||||||||||^|r-.|||,-l— .MrHCOr-.^|||||||||||| n ^ 1 1 i 1 1 1 1 1 |--I(m| l^r-l^rHrM^rtrHIM^I |^r-lj | | | | 1 1 1 I-- "o 1 1 1 1 1 1 1 1 1^1 1 |co|Tt<.— 0^^rttNCC.-c.-H| MI 1 1 1 1 1 j 1 CT 1 1 1 1 l^-l MI |^«oqoc5(Moq(NrH| | | | | | | | | 1 | | « 1 I 1 1 l'~'l 1 |^'>l[f?^"*'MG^»ß'^^|'-HTOCN|r— 1| 1 1 1 1 1 j 1 1 1 00 ^ 1 1 1 1 I 1 1 j<-'(Mr-l.-lf-(^CCJ'^l-HCC'*^^5^||-.||||||||^|||||!|||| o 30 IMMMI"MMI-MMMMMMMMMM lO M M M M-- M M M M M M M 1 M M M M M CO i M 1 - - M - - M M M M M M M M M M M M 1 ^ o M M - - M 1 M M M M M M M M M M M M M oq ^ C-. Ci '-i ;^( '0 ■^'^^ *o i". cc Ci c; »-^ ^j ^0 ^^»o ;o *- '^ ^^) o '-H <^i ^ *^»-.> :o *^ oo c^ Ci ---i '^^^t ^ »-< ^i (5i Sl ^i i^i ©i i?i Oi (^ ^i ^ So 30 30 ^ 20 20 ^ =-: ^o -Ä^ -cf -^ ^ *q. -4 -^ *^ -<^ *^ U-> Is^ \^ 'Jl e2 ■ßuaui'Bqg 21—2 1G4 Cooperatlve Invcstigations on Plauts 1 ' ' ' ' 1 ' — « IN O« K l- re 50 (N (N •* (N (N — — ^ 1 1 1 1 1 1 1 ^^ 1^ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ^ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 **'i 1 1 1 > 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -H 1 1 3" 1 1 1 1 III 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 •^ 1 1 1 1 1 1 1 1 1 1 1 i 1 1 1 1 1 1 1 1 1 1 1 1 1 'l 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 >- 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 •::_ ^ 1 1 i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -l 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i 1 1 1 1 1 1 1 1 1 1 1 1 ^ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 lllllllllllllllll||||||||||||^^||||(N|||||||||||ll ^ ^ IIMlJllllllllllllllllll^JIIIIJn-l^llllllllllllllll •M 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ^ 1 1 ^ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ^ Illllllllllllllllilllllllllll^n-'IIIIIIIIIIIIIIIII 4 lllllllllllllllllll^llllll|ra|--ll!-H|^|„||lllllllll X lllllllllllllllllllllllll'N|'-<'-l|r-<^>^||,-l||||||||||||| X C^. lllllllllllllllllllllllllll-l|||||||||||||||||| X IIIIIIIIIIIIIIIIIIIII^II'-<|IM|>-4-l||l||{||llllllllll 1 1 1 1 1 1 1 t 1 1 1 |i-H|i-i|'^^(MrM:o^74^(N^I*tl |^|f-*FH| [ 1 1 t 1 I 1 1 1 1 1 1 1 1 ^ 1 1 1 1 1 1 1 1^1 1 1 |t-»^10IC0-^— ^^NO'^MINCT'NI 1 Ir^l 1 1 1 1 1 1 1 ! 1 1 1 1 1 1 1 1 1 |ll|ll|lll|(M|^|>-:C0OW(Nr- irHr-lrHr-ll— || lllllllll 33 X 2| 1 1 1 1 1 1 1 1 1 1 1 |i-<l''^-«l»|rH|rt|I|||||||||||||||||||,|| ^ llllill|i|lll>-i|rt-4.-i-«in^F-i||(N||||||||iiii|||||{||||| o CS llllll||||0)(N»i-<~4 thu coldur ul' tho coloured patchcs. Class a = ycllow. CUiss b = fiiwn yellow. Class c=palc wild coloui- Cla^s rf = dark wild cülour| Class e = black, aiid Class f= " lilac " = pale bluu gray ; at inv.suut uiily uxliibilud hy Uie oti'spriiig of liybrids. Il, is not suggested that these form oiiu eolour series. Table I. shews Ihe distribution of eolour and the colotir itsclf in all tlie individuals of twenty faniilics [)roduccd by crossing waltziiig with albinu nüee. All the niice in thc Table have black cycs. TABLE 1. Number of Gross Group I. • Group II. Group III. Group IV. Group V. Group VI. Totais a b c d — _ _ e _ a b c j d e 1 1 _ _ _ b c 2 1 2 2 1 2 2 I 1 I "~ 2 2 3 2 3 d 1 1 2 1 e _ 1 2 a b _ 1 1 6 I 2 I d 2 1 e 1 a \b c d e a b c d e i ii vii viü ix xii xiii xvi XX xxvii* xxxiii* XXXV xlv lix* Ixi Ixii Ixxiii Ixxiv Ixxv Ixxvii Ixxviii Ixxxii* Ixxxiv Ixxxv Ixxxvi xciv* c ci ciü - - 2 1 1 2 5 1 1 4 1 1 2 1 1 1 1 1 1 1 2 1 2 5 2 6 1 I 4 3 4 4 1 4 5 5 5 4 3 1 2 1 3 1 1 1 ~ - 1 _ 6 1 7 6 6 ö 4 3 6 ] 1 6 4 2 7 6 6 7 h 8 4 2 9 7 7 2 9 9 8 Totais - - 3 - - - - 21 7 2 27 5 3 - - 12 3 1 7 49 10 3 - 1 - 154 * In these crosses the parents werc, $ waltzing mousc x J albino ; in all the rest the parcnts wcrc ? albino x j waltzer. 168 !Second Report oii Croas-bred MIce Froni tlie i'ulluwin«^ re-classiKcatioii of the a,bovc Tablo I. thc t'ro(iuency of degrees of whitcuuss aniong 15-i hybrids will bc at onco cvidcul. Group 1 3 Group 2 30 Group 3 35 Group 4 16 Group 5 (J9 Grunp 6 1 154 That i.s to say thc majority of hybritls havu coiüurcd patchcs in, roughly, the saiiic place as thc waltzcrs liavc ; or are wild-colourcd rnicc cxcept for thc belly. Thc foUowing arrangeincnt shcws thc frcquency of colours among 154 hybrids. Class a 7 Class b Ckiss c 112 Clana d 2(3 Class e 9 154 That is to say by far tlio grcatcr uumbcr of hybrids cxhibit wild rolnur. Offspriiif/ of Hi/brids. Hybrid.s liavc bccii paiicd inter sc and crosscd with albiuos. A. Off spring of pairs of Ili/brids. Tiio numbcr of mice which waltz in this gcneration is eight. At picsent nine albinos have appearcd : thc actual arrangemcnt of thc colours and the colovirs themselves arc hcrc shewn : thc Icttcr w indicatcs that thc niousc cxhibits waltzing niovciiients, and p that it has pink cycs. It is uniicccssary to say that albinos havc pink cyes. //,. ? 2rf X / 2c. 3 Young. %f(wp), 2a (p), 2b. 11,. ? Ic X ^ 2c. 4 Young. 36 {p), 3c, 4a {p), 4« {p). H.,. ? 2rf X J- 3c. 5 Young. Sa, 36, 36, 3e(w), Sf. II„. ? 2c X c/- 2c. 7 Young. 4 albinos, 'db(wp), '6b (p), 3c. A. D. Darbishire 169 H,,. ? 2c X ^ :5c. 7 Young. 2 alhinos, 2c, 2c, 2c, 2c (?ü), 4e (?y). H.,,. ? 2c X ^ 2c. () Young. Albino. Albino (tv), 2c {^u), Sc {tu), 2e, 2/. lU % icx^ 3c. 5 Young. Albino, -e, 2c, 4e, 4d That is to say there are .S7 young froiii 14 hybrid parents of degrccs of whiteness incliuled in Group.s 2 and 3. The young may be classificd according to degrees of whiteness and according to their colour. Degrecs of Wliitenesa Nature of Colour Albinos 9 Albinos 9 Oroup 1 Glass a 4 Qroup 2 12 Glass b G Group '?> 11 Glass c 9 Group 4 5 Glass d 1 Glass e 5 37 Glass f 3 37 B. Offspring of Hybrids and Albinos. The following is a record of young produced by crossing a hybrid vvith an albino. Hi. ? albino x (/" 3c. 4 Young. 5c, 5c, .5c, 5c. i/o. $ albino x J' 3c. 9 Young. 5 albinos and 4e, id, -id, 5d. Hf. $ albino x ^ 5c. 5 Young. 2 albinos and 5c, 5c/, Qe. H^. % albino x ^ 5c. 5 Young. 1 albino and Qd, Qd, Qd, 6c. H(,. % albino x ^ 5c. 4 Young. 2 albinos and 6d, 6d. Hn. ? albino x / M. 6 Young. 5 albinos and Ge. H,„. ? 2c X A.. + ^-l^-)i^)AX- (3). An alternative formula, in which central differences are used, is that recently given* by Professor Everett. When we do not require to go beyond third differences, this formula is most conveniently written u = «„ + ÖA«o - -^^^ — ^ S'"i - ^^ g/ ^ S"«o (4), where S'wi and B-tio are the second central differences of u^ and Wj (i.e. B-Ui = A'u^, 8^0 = A-(/_i), aud (/> = 1 - Ö. In practice, when second and higher differences have to be considered, and a mechanical method of porforming multiplications is available (e.g. by means of a Brunsviga), it is more convenient to use differential coefficients. Let the means of the pairs of odd differences above and below the line of ii„ be denoted by /j,Su„, fiS'iio, ... (i.e. /^8(^, = -^ (Ai(„ + A<<_i), /u.S'it„ = |(A''i6_i + A'fLj), ...), and let Dt(o, B-Ufi, IPuii, ... be the values for x = x^ of the successive differential coefficients of It with regard to x; theuf, 8^ having the same raeaning as in (4), IiDua = /i8»„ - ^ fiBhi^ + ^Vm^^^'o - •■•\ hWui) = S-u„ — f\B*u„ + ... hWUa = flBhlo-^/jl,B''Uf, + ... )■ (5), h*D^Uo = B%- ... etc. and, for x = Xo + dh, u= «0 + [hDu„ + \e [h'D-u, + {6 (hWiio + ...)]] (6). * Journal of the Institute of Aetuaries, Vol. xxxv. p. 452. t Proe. of the Land. Math. Soc. Vol. xxxi. p. 465. 17(3 New Tables of the PruhahUitij Inlajral We need only use inside the square brackets the numerical valucs of Ö and of /ii)«„, h-iy-Ut,, ..., if we put tlie sigu of 6 JiDu^ outside, and follow the rule of signs inside. The rule is that a minus sign niust be iiiserted, if 6 be positive, when- ever there is a change of sign in the scries hDu^, h^D'u^ or, if ö be negative, whenever there is no change of sign in this series. Thus, in Table I., taking u to be ^ (1 + a), we havc for a-o="40 (omitting decimal point) 3G826i -147 -3 hüll, h-D-u, h'I?Uo 3G827 -147 -3 and thcrefore, for a;="40+01ö, i (1+ a) = 10-- (6554217 + 6 [36827 - \d (147 + 'Ö . 3)]). For a;=-4ü--01öwe should replace +6, -W, +16, hy -0, +\e, -^0. The formulffi giveu by (3) and (4) would be, for a; = '40 + 'Olö, Hl + «) = 10- (6554217 + 36753(? + 150 ^^^ - 3 ^^i:^^^?^^) , Hl + a)= 10-' (6554217 + 36753Ö + 150 ^JlZL^ + 147 t(k:J^)\ . There is no difllculty about tlie divisions represented by the coefficients i, ^, ... of in (6); but, if wc wish to avoid thcm, we may calculate lv'D^u^j2\, h^Ifiu^/3 !, ... and write the forniula u = /*o + Ö [hDu, + (H=^=«o + QJi'LPuc, + ...)!] (7). 3. Inverse Interpolation. In most cases N^ and Nn [= iiV"(l + «) and ^N{\ - a)] are known, and we requirc x. If Hl + ") ''^^'•^ Hl ~ '*) ^^^ both less than 90, we can use Table III., which givcs x in terms of a = {Ni — N^jN. But, if either ^(1 + a) or Hl~*) l'ß greater than 'OO, we niust use Table L, by inverse Interpolation. By (6) we have, if a; = a;, + dh, e = {u- »„) - [hDu, + hß {IrD'u, + W {h'D'Uo + ...))] (8). or, if x = Xo— 0h, = (u„-ii)^[hDu„-^d [k''I)^Uo-^0(hWit„-...yf\ (8a). The value of 6, and thence that of x, is obtained by successive appro.ximations. Suppose, for instance, that Hl + a) = -654. If .r = •40-01 ö, we have from Table I. ö= 14217 H- [36827 + iö (147 - 0\]. W. F. SllEl'PARD 177 A first approxiination gives = 14217^36827 =-38605; and, with this value, the correctcd divisor becomes 36855-2, which give.s f'or a second approxiination = -38070, and therefore « = •39614.2.5. The conoct valuc, as givcii hy Table III., is ,6 = -3961424. The degree of accuracy with which x can be obtained by this method depends on thc relation of maguitude between the differences of x and of ii. In the above example, to a difference of 'Ol in x there conesponds a difference of veiy little more than one-third of Ol in ^(1+a); and therefore, if *• is calculated from Table I. to seven places of decinials, it will only be accurate within about 2 in the lastfigure. The possible inaccuracy of *• increases as ^{1 + a) increases. But this is not importaut, as the " probable error " of oc, for any given minibcr of observations, also increases. 4. Smoothiiig. In arraiiging a table, with difierences, for the calciilation of any ciuaiitity v, it is usual to enter in the iliti'crence-cohimns the actual (or " tabiilar " ) diticrences of thc vahies of u as tabidated. In the prosent tablcs I have adopted a different method, and ha\-e given the diflferences as near as possible to the differences of the true values of u. The object of thi.s is to enable greater accuracy to be obtained when required. If we only want u to tive or six place.s of decinials, it is imniaterial whether we use the tabular or the corrected dift'orences. But, if we wi.sh to have it as accurate as possible, we can alter the tabulated values by inspection. Looking, for instance, at the commencenient of Table L, it is clear that the tabulated values of i(l+a) are too great for .);='01 and .r=-03, while they are too small for .r=-02 and .r = -04. Taking M=i (1 -\-a) x 10", so as to omit the decimal point, the table may be written X V, A + A2 ■00 5000000 39894 -ö ■Ol 5039894-0 39890-(l-ö-(^) 4 + (l-2ö-(#)) ■02 5079783 + <^ 39882 -((^ + x) 8 - (1 - ö - 2<^ - ;() ■03 5119665 -;( 39870- (1-x-V') 12 + (l-<^-2;(-<|/) ■Ok 51 59534 + i|r By tabulating v, by differences of -05 or -10 in .r, it will be found* that the third difference in Table L, for these values of .r, is almost exactly 4. We see therefore that ö + c^ and x + ^ are both greater than i, and (^+x is less than \; while ö, \-6-, 4> + x^ ^^'^ ^~X~^ ^rs all very nearly equal. The values ö=-4, <^ = -2, j( = % V'=^i satisfy the.se conditions ; and, as a matter of fact, they give for i (1 + a) values which are correct within 1 x lO^**. * For the relations between diflferences of w for large and small differences in .r, see Proc. I.nnd. Muth. Soc. Vol. XXXI. pp. 46S— 471. Biometriliii ii '.^3 178 New Tables uf tlie l'rubabilitn /uta/ral Examples of Application. 5. Expression of data in terms of x. Whcu a distribution is nearly normal, we may statc tlic data by cxpressing A' (soc § 1) in terms of x. For an exaniple, take tho head-brcadths of 3000 criminals, givun on p. 214 of BiometHka, Vul. I. The interval iu X is '1 of a ceutimetre : but, for brevity, we shall take intervals of •3 of a centimetre. We sliould first note tbc probable eiTore. If, of the N values, i\'', lie below A' and iVj above it, thc probable error in A'i or iV„ is + Q '^^■N'i^'JN, where Q = '67449 ; i.e. it is an eveu cliauce that N tiines the truc pruportion of values below A' lies between N,- Q'. E. is ± Q \/479 X 2.521 -f- 3000 = 13*.5. Calculating thc probable crrors, the data may be expressed thus: — A' -\'i X, P. E. 1 A' ^Vi .V, P. E. 13-35 3000 15-1,5 2364 636 + 15-1 13-65 8 2992 + 1-9 ; 15-7Ö 2763 237 + 100 13-95 38 2962 + 4-1 16-05 2925 75 + 5-8 H-25 l.'JV 2843 + 8-2 16-35 2978 22 + 3-2 14-55 479 2.521 + 13-5 16-65 2997 3 ± 1-2 U-S5 1077 1923 ±17-7 16-95 3000 — 15-13 1762 1238 + 18-2 1 Now calculatc the values of x and of z correspouding to A', = ^A'^(l + ot). An error of 6 in x is equivalent to an error of zd iu ^ (1 + a), so that the above values of the P. E. have to bc diviilcd by 30005 to give the P. E. iu x. We thus get the data in the form : — .Y X- P. E. X X P.E. 13-35 — X 15-45 + -800 + 017 13-65 -2-786 + -077 15-75 + 1-412 + -1 )23 13-95 -2-236 ■t--042 16-05 + 1-960 + -033 U-25 -1-625 + -020 1 16-35 + 2-441 + •052 14-55 - -996 •f-019 16-65 + 3090 ±-135 ■14-85 - -361 H-OU) \ 16-95 + ac — 15-15 + -221 ±016 This, it shoidd bc obscrvcd, is mcrely a statcment of facts, and docs not involvc any assumption as to the distribution being really normal. 6. Interpolation*. By means of these values of x, we can intcrpolate for values of A lying towards tho extremities of thc ränge, where thc differences * For a fuUer discusgion of the methods employed in this and thc ncxt two sections, Bee Journal 0/ the Royal Statigtical Society, Vol. LXiii. pp. 433 — 451. W. F. SlIKPPARD 179 of J.V, or iV.j arc nsiiully irregulär. Thus, calculating tlie valiu-s uf ,r für A' = 14-05 anil 141.') by ineans of tlio first ditilbreiico alonc, aiul thence cak-ulatiiig i ( l + «) froni Table I. (remeinberiiig tliat x und a are negative), wo get the l'ulluwiag results, as compared with the actual obscrvations : — ,\' x A(l-n) A(l + «) N^ = \N{\+a) Calculated vahie Actual valiK" 14-or, iJt-ir, - 2-032 - 1-829 -9789 -9663 -0211 ■0337 63 101 61 97 It is (jiiite possiblo tliat the discrepancy between the ealculati'd and the actual values is mainly due to the errors of random selection of the 11 8 individuals lying between X = 13-95 and X = 14-25. 7. Certain Special Cases. The mcthod is especially uscful (r/) where the difterence.s in A' are irregulär or are large in comparison with the Standard deviation, and (6) in dealing with tho " arrays " in cases of normal or nearly normal correlation. As an example of the former, Prof. Pearson has jirovided nie with the following rcsnlts obtained by Miss C. D. Fawoett. Muttlimj of Mimulus Liitens. A' = Numbei- of splotches Less thau 50 öO to Ol i'yi to 71 Number of individuals 18 43 Hl More than 71 5G Total 204 This eives three values of *■ in terms of A', viz : — X 49^ Gl^ X - 1-352 + ■084 - -527 + 002 + -599 ± -ÜG3 The differenccs in .r are -825 and 11 2G, whereas the differences in A' are in the i-atio of 6 : 5. Having regard to the probable errors, it is very doubtful whether the distribution can be treated as normal. If it can be, the truo values of .-« may be somewhat as follows : — A 49i x (corrected) — 1-503 Ratio of correction to P. E. — 18 These would give a meaii of GGGl, and a Standard deviation of ir05, the probable errors being re-spectively + 52 antl + -37. 23—2 6U 71i -■417 + -488 + 1-8 -1-8 18U Netv Tahles <>f thc /'robahl/K;/ htln/Kd This, ot' courso, is guesswoik. If thero are inore thaii three or four valiies of x tu be dealt with, we can do our guessing by graphic methods. The values of X should be plotted as oidinates, with a mark on each side to shew the probable error: and we have then to draw a straight line which sliall make the error as sniall as possiblo, allowance being made for tbe difterout valiies of the probable error in the differeiit ordinates so plottuii. It niust be reinembered that the crrors are not independent, but eorrelated : the correhition between the crrors in any pair of values of a; being positive. If we want to procccd more rigidly, we miist use the actual values of x, and take account of diff'eronces. This is practically equivalent to regardiug X as an unknown function of some other quantity Y, whose values are normally distri- btited ; the rclation between X and 1" being such that dX'dV is always positive. 8. Ctilculation of ordinates. The orilinate Z of the curve of frequency is dXJdX. In most cascs this can be calculated directly froin the data, by the first formula of (5) (takiiig « = J',) or a siniilar formula. Whcre the difiFerences in u are too irregulär for this, bvit the diflFerences in .r are comparatively regiilai-, we can u.se the data in the form shewn in § 5 : we have then Z= X.d^ (1 +a)ld.v. dsldX=Nz . dx/dX. Thc values of rf.i dX aro. given by the data, and the value.s of » are found froni Table II. (t. TestiiKj for normal didrihutiun. This is the purjjose for which tables such as Table I. are most froijuently eniployed. It is not necessary to give any examplcs here. 10. Galculation oj correlation-volumen. The formula, given by Professor Pearson*, for calculating the double-integral cxpressing normal correlation, in- volves a factor _e-S(x- +!/'') 27r which is the product of the values of z as given by Table II. for .<• and for y rcspectively, and therefore is easily fonnd from thnt Table. Construction of tlie Tables. 11. The tabulated values are all given to seven dcciuial places (and, in the lattcr {wirt of Tables I. and IL, to ten det-inial place.s). They were originally calculated to two or thrcc uiore placas, the final figure.s being then corrected. Where the tinal figure Wfus doubtful, the value was calculated specially. The dirt'erences are taken froni the largcr table, the last figure being corrected : but doubtful values were not .sjiecially calculated. For constructing Table.s I. and II. up to :r=2'5(), the value.s of c were found for the inter- uicdiate values -OGö, 'Olö, 'OSO... of .r by successive luultiplication ; each tentli value being checked by Newniau's table t of e - *. The diflereuce.i being then ttvken, the vjxlues of i ( 1 + a) for • Phil. Tram, series A, Vol. I!t5, pp. 1 — il. t Cumh. l'hil. Suc. Tram. Vol. xin. l't. :t. W. I'\ SllKI'l'ARD 181 .r='01, '02 ... worc obtaiiied by quailraturc*, .■uiil t.hose of : by iiiterpolatioii. Fdi- tlie rcinuiiider of the twü tables, the values of z wero found iVoni thoso of log,,, 2, which are easdy calculated, and thence the vahie.s of ^(1+«) were obtained by miadratui-o. For checkiiig tbe table of i(l+a), values were directly calcidated at intervalst: and liotb t,d)Ie.s were furthi-i- checkcd by the cakidations requirod where a fhial figuro was doubtful. l'"or constructing Tables III. and I\'., the vahie.s were iirst ulitained appro.Kiuiatcly tu .suven place.s : and the tables were then e.xtended, by a method explained el.sewhere J. The extension gave s to nine and .r to eleven place.s. The table was eheeked by direct cakiilition for h = 'I, "2, '3.... * See Piüc. of Luntl. Math. Svc. Vol. xxxi. pp. 179—482. t Some use was also made of Burgess's table» {Truns. Roy. Soc. Edin., Vol. xxxix., Pt. 2, No. 0), in which a is given (to alargenumber of figures) in terms off = x/V2. But they were only useil inciclfntally aud the two sets of tables may be regarded as independeiitly calculated. J Prof. 0/ Loiul. Math. Soc. Vol. xxxi. pp. 123, 43',l. TABLES I. AND II. Area und Ordinate in terms of Äbscissa. Note. For values of the abscLssa x from '00 to 4'.50, the vahies of the arca ^ (1 + a) and of the ordinate z are given to 7 deciinal places (pp. 182-7). For values of x from 4'50 to 6'00, the values of | (1 + a) and of 2 are given to teu decinial places (p. 188), but the initial figures are omitted. Hence, in using this latter portion of the tables the figures in the column for ^(1+a) must have OODÖO prefixod, aud those in the column for z must have sufficient zeros prefixed to bring up the total of decimal figures to ten. For exaniple, against ic = 5"7ö we have 99955 and 264. but we niu.st read h {l + a) = •9991)91)9955 and s = •0000000204. 182 Neil- Tahles of the Prohahilitn /nteijrul TABLES I. AND II. Area aml Ordinate in terms of Äbscissa. i(\+a) ■00 ■Ol ■0:.' ■OS ■04 ■Oö ■00 ■07 ■08 ■09 ■10 ■11 ■li •13 ■15 ■10 ■17 ■IS ■19 ■M ■^3 ■26 ■27 •28 ■29 ■30 •31 ■32 ••'■'/ ■35 •SG •37 ■38 ■30 ■1,1} ■41 ■1,2 ■1,3 U ■J,5 ■1,6 ■J,7 ■1,8 ■J,'J ■HO ■nOOOCKK) ■.JU7'J78:i •5119605 •5159534 ■5199388 ■5-23922:2 •5279032 •5318814 •5358564 ■5398278 ■5437953 •5477584 •5517168 •55567(X) •559G177 ■5635595 •5674949 •5714237 ■5753454 •5792507 •5831662 ■5870644 ■5909541 •5948349 •5987063 •6025681 •6064199 •6102012 ■6140919 •6179114 •6217195 •6255158 •(1293(JOO •6330717 •6368307 ■6405764 •6443088 ■6480273 ■6517317 •6554217 •6590970 •6627573 •6664022 •67003 14 •6736448 •6772419 •6HU.H225 •6^43.HG3 •6879331 •6914625 A + 39894 39890 39882 39870 39854 39834 39810 39782 39750 39714 39675 39631 39584 39532 39477 39418 39355 39288 39217 39143 39065 38983 38897 38808 38715 38618 38518 38414 38306 38195 38081 37963 37842 37717 37589 37458 37323 37185 37044 36900 36753 36602 36449 36293 36133 35971 35806 35638 35467 35294 d; 21 28 32 36 40 44 48 51 55 59 63 67 71 74 78 82 86 89 93 97 100 104 107 111 111 118 121 125 128 iSl 135 138 141 144 147 150 153 156 159 162 165 16S 171 173 176 •3989423 •3989223 •3988625 •398762S •3986233 •3984439 •3982248 •3979661 •3976677 •3973298 •3969525 •39G5360 •39608(J2 •3955854 •39505 1 7 ■3944793 •3938684 ■3932190 ■3925315 •3918060 •3910427 •3902419 •3894038 ■3885286 •3876166 •3866681 •3856834 ■3846627 ■3836063 •3825146 ■3813878 •3802264 •3790305 •3778007 ■3765372 ■3752403 ■3739106 ■3725483 •3711539 ■3697277 ■3682701 ■3667817 ■3052027 •3637130 •3021349 ■3005270 ■3588903 ■3572253 •3555325 ■3538124 •3520653 199 598 997 1395 1793 2191 2588 2984 3379 3773 4166 4558 4948 5337 5724 6110 6493 6875 7255 7633 8008 8381 8752 9120 9485 9847 10207 10561 10917 11268 11615 11958 12298 12635 12908 13297 13623 13941 14262 14575 14885 15190 15491 15787 10079 16367 16650 16928 17202 17470 A- X ~ 399 ■50 399 ■51 399 ■52 398 ■53 398 •54 397 ■55 397 ■56 396 ■57 1 395 ■58 394 •59 393 ■CO 392 ■Gl 390 ■62 389 ■6S 387 •64 386 ■GS 384 •66 382 •67 380 •68 378 •69 375 •70 373 •71 371 •72 368 ■73 365 ■r4 362 •75 360 ■76 357 •77 354 •78 350 ■79 347 ■SO 344 •81 340 •82 337 ■83 333 •84 329 ■85 325 ■86 322 ■87 318 •88 313 ■89 309 ■90 305 ■91 ' 301 •92 290 •93 292 ■94 288 ■95 283 ■96 278 ■97 274 ■98 269 ■'.19 264 l-iiil 4(1+«) ■6914625 •6949743 •6984682 •701944U •7054015 •7088403 •7122603 ■7156612 ■7190427 •7224047 •7257469 •7290691 •7323711 •7356527 •7389137 •7421539 •7453731 ■7485711 •7517478 •7549029 •7580363 •7611479 •7642375 •7673049 •7703500 •7733726 •7763727 •7793501 •7823046 •7852361 •7881446 •7910299 •7938919 •7967306 •7995458 •8023375 •8051055 •8078498 •8105703 •8132671 •8159399 •8185887 •8212136 •8238145 •8263912 •8289439 •8314724 •8339768 •8364509 •8389129 •8413147 A + 35118 34939 34758 34574 34388 34200 34009 33815 33620 33422 33222 33020 32816 32610 32402 32192 319K1 31707 31551 31334 31116 30896 30674 30451 30220 30001 29773 29545 29316 29085 28853 28620 28387 28152 27917 27680 27443 27205 26967 26728 26489 2()249 2(i008 25768 25527 25285 25044 24802 24560 24318 W. F. SUKITAIII) 'l'AüLKS I. AND II. — {continued). J83 ■;j.'i20(ir):{ •:ir)029U) ■:i4«4l)2ö ■31(i(l(!77 •3448180 •3429439 •3410458 ■3391243 •3371799 •33.")2132 •3332240 ■3312147 •3291840 •3271330 •32Ö0G23 •3229724 •3208638 •3187371 •31fi:i929 •3144317 •3122039 •3100003 •3078513 •3050274 •3033893 •3011374 •2988724 •2905948 •2943050 •2920038 ■2896910 •2873089 •2850364 •2S2G945 •2803438 •2779849 •2750182 ■2732444 •2708040 •2084774 •2600852 •2630880 •2012803 •2588805 •2504713 •2540591 ■2510443 •2492277 •2408095 ■2443904 •2419707 17734 17994 18248 18497 18741 18981 19215 19444 19607 19880 20099 20307 20510 20707 20899 21086 21267 21442 21613 21777 21936 22090 22239 22381 22519 22050 22777 22897 23013 23122 23227 23325 23419 23507 23589 23006 23738 23805 23860 23922 23972 24017 24058 24093 24122 24147 24107 24182 24191 24196 A^ 264 259 254 249 244 239 •231 229 224 219 213 208 203 197 192 187 181 176 170 105 159 154 148 143 137 132 126 121 115 110 104 99 93 88 83 72 66 Ol 50 51 45 40 35 30 25 20 15 10 l-IHt l-ll.' i-ii.: i-d.-, I IIC i-ii: l-IO I-II l-ht 1-n 1-15 1-tfi 1-17 1-IS l-l'J 1:.'0 i-:n !■.'.' 1;.'.J 1-24 l-:^ß 1-26 1-27 1-2S 1-20 1-30 ]:V 1-S2 1-SS l:U 1:15 rr.i; 1-37 1-SS 1-30 1-40 1-41 1-42 }-43 1-44 i-4r, 1-46 '■47 1-4S 1-40 l-öO ■8413447 •8437524 ■8461358 ■8184950 •8508300 ■8531409 •8554277 •8570903 •8599289 •8621431 •8043339 •8005005 •8680431 •8707019 •8728508 •8749281 •8709756 •8789995 •8809999 •8829708 ■8849303 •8868600 •8887076 •8900514 •8925123 •8943502 •8901653 •8979577 ■8997274 •9014747 •9031995 •9049021 ■0065825 •9082409 •9098773 ■9114920 ■9130850 •914(i5G5 •9102007 •9177350 •9192433 •9207302 •9221902 ■9236415 ■9250663 •9264707 •9278550 ■9292191 ■9305634 ■9318879 •9331928 A + 24070 23834 23592 2335 1 23109 22808 22020 22380 22145 21905 21605 21420 21188 20950 20712 20475 20239 20004 19709 19535 19302 19070 18839 18000 18379 18151 17;)24 17097 17472 17248 17020 10804 16584 10365 16147 15930 15715 15501 15289 15078 14868 14600 14453 14248 14044 13842 13042 13443 13245 13049 242 242 242 242 242 211 241 211 240 240 210 239 239 238 237 237 230 235 235 234 233 232 231 230 229 228 227 22() 225 224 223 222 220 219 218 217 215 214 212 211 210 208 207 205 204 202 201 199 197 190 194 ■2419707 ■2395511 ■2371320 •2347138 ■2322970 ■2298821 •22716!K! •2250599 ■2220535 •2202508 •2178522 •2154582 •2130091 ■2106850 ■2083078 ■2059363 ■2035714 •2012135 •1!)8S(;31 ■1:m;52o5 ■1941861 •1918002 •1895432 •1872354 •1849373 ■1826491 •1803712 •1781038 •1758474 •1736022 •1713086 •1691468 •1069370 •1647397 •1625551 •1603833 •1582248 •1500707 ■1539483 •1518308 •1497275 ■1476385 ■1455641 •1435046 •1414000 •1394300 ■] 374165 •1354181 ■1334353 ■1314684 ■1295176 24190 24191 21182 24168 24149 24125 24097 24004 24027 23986 23940 23890 23836 23778 23715 23649 23578 23504 23426 23344 23259 23170 23077 22981 22882 22779 22073 22504 22452 22337 22218 22097 21973 21847 21717 21585 21451 21314 21175 21033 20890 20744 20596 20440 20294 20140 19985 19828 19069 19508 A- + 10 14 19 24 28 33 37 41 40 50 54 58 62 60 70 74 78 82 85 89 93 90 99 103 100 109 112 115 118 121 124 127 129 132 134 137 139 142 144 146 148 150 152 154 155 157 159 160 162 1S4 Neiv Tables of the Probabllitif Integral TABLES I. AND \l.—{continued). i .1 -t". •0331028 •0344783 ■03:)744:) •!l3(i09I6 •03^:2198 •0304202 •0406201 •0417024 •042i»466 ■044082(5 ■04.")2()()7 •04ß301 1 •0473S39 ■0484493 ■0404074 •9505285 ■051542.S ■0525403 •0535213 ■0544800 ■0554345 •9563671 •0572838 •0581849 ■05!)O70n ■9500408 •0607961 ■0616364 ■0624620 ■0(>32730 ■9640697 ■9648521 ■9656205 ■9663750 ■9671159 ■0678432 ■06S.-..-)72 ■9692581 ■9600 160 ■07W>210 ■0712834 •9719334 •0725711 ■9731966 ■973K102 ■9741110 I ■075f)021 I ■075580H I ■0761482 ■9767045 ■0772400 A + 12855 12662 12471 12282 12094 11908 11724 11541 11360 11181 111104 10828 10654 10482 10311 10142 9975 9810 9647 9485 0325 0167 9011 8856 8704 8553 8403 8256 8110 7966 7824 7684 7545 7409 7273 7140 7009 6879 6751 6624 6500 6377 6255 6136 6018 5902 5787 5674 5563 5453 194 193 191 ISO 1M8 186 184 183 181 179 177 176 174 172 170 169 167 165 163 162 160 15S 1 5(; I 55 1 53 151 1 III I 17 1 16 1 14 1 12 140 139 137 1 35 133 132 130 128 126 125 123 121 120 118 116 115 113 111 110 ins ■1295176 •1275830 ■1256646 ■1237628 ■1218775 •1200000 ■1181573 ■1163225 ■1145048 •1127042 ■1 109208 ■1091548 ■1074061 •1056748 ■1039611 •1022649 •1005864 ■0989255 ■0972823 •0956568 ■0940491 ■0924591 •090S870 •0803326 •0877961 •0862773 •0847764 •0832932 •OS 18278 ■0803801 •0789502 •0775379 •0761433 •0747663 •0734068 •0720649 •0707404 •0(i94333 •0681436 •0668711 •0656158 •0643777 •0631566 •0619524 •0607652 •0595947 •0584409 ■0573038 •0561831 •0550780 ■u.-i3001O 10346 10183 1901H 18853 18685 18517 18348 18177 18006 17834 17661 17487 17312 17137 16962 16786 16609 16432 16255 16077 15899 15722 15544 15366 15188 15010 14832 14654 14477 14300 14123 13946 13770 13594 13419 13245 1 307 1 1 2897 1 2725 1 2553 12382 12211 12041 11873 11705 11538 11372 11206 11042 10879 + 162 163 165 166 167 168 169 170 171 172 173 174 174 175 176 176 177 177 177 178 178 178 178 178 178 178 178 178 177 177 177 176 176 176 175 175 174 173 173 172 171 170 170 169 168 167 166 165 164 163 162 S-00 2-(n 2-0:> äOJ, 2-U5 2-06 2-07 2-OS 2-00 210 211 2-12 2-1.3 2-lJ, 2-ir, 2-16 2-17 2-lS 2-1!) 2'2t) 2-21 3'22 2-2S 2-24 2-25 2-2r. 2'27 2'2S 2-29 2-SO 2:11 2-32 2-33 2-3i 2-35 2-36 2-37 2-3S 2-30 2-J,0 2-41 2-J,2 24.1 2JiJ, 2-40 S-J,6 2-1,7 2-l,S 2-1,0 2-50 *0+«) •9772499 ■9777844 •0783083 •078S217 ■0703248 •9798178 ■9S030O7 ■9807738 ■0812372 ■0816911 •9821356 •0825708 ■9829970 ■9834142 •9838226 •9842224 •9846137 •9849966 ■9853713 ■9857370 ■0860966 •9864474 ■9867906 ■9871263 ■0874545 •0877755 •9880894 •0883962 ■0886962 ■0889803 •0802759 •9895559 ■0898296 ■0000069 ■0003581 ■9906133 ■9008625 ■0011060 ■0013437 ■991575s ■9918025 ■002O237 ■0022307 ■0024506 ■0026564 ■0028572 ■0030531 '.1032443 ■0034309 •9036128 •9937903 A + 5345 5239 5134 5031 4929 4829 4731 4634 4539 4445 4352 4262 4172 4084 3998 3913 3829 3747 3666 3587 3509 3432 3357 3283 3210 3138 3068 2999 2932 2865 2800 2736 2674 2612 2552 2492 2434 2377 2321 2267 2213 2160 2108 2058 2008 1960 1912 1865 1820 1775 48 47 46 45 44 W. F. SlIKPPARI) 185 TABLES I. ANU 11. — (continued). ■0.-)39910 •0529192 ■0518630 ■0508239 ■0498001 ■0487920 ■047799G ■0468226 ■0458611 ■0449148 •0439836 ■0430674 ■0421661 ■0412795 ■0404076 ■0395500 •0387069 ■0378779 ■0370629 ■0362619 ■0354746 ■0347009 ■0339408 ■0331939 ■0324603 ■0317397 ■0310319 ■0303370 ■0296546 ■0289847 ■0283270 •0276816 •0270481 ■0264265 •0258166 •0252182 •0246313 ■0240556 ■0234910 •0229374 •0223945 •0218624 •0213407 •0208294 •0203284 •0198374 •0193563 •0188850 •0184233 •0179711 •0175283 10717 10557 10397 10238 lOOSI 9924 9769 9616 9463 9312 9162 9013 8866 8720 8575 8432 8290 8149 8010 7873 7737 7602 7468 7337 7206 7077 6950 6824 6699 6576 6455 6335 6216 6099 5984 5870 5757 5646 553S 5428 5322 5217 5113 5011 4910 4811 4713 4617 4522 4428 A2 + X 162 2:50 161 ;>-.-,l 160 2;',J 159 2-n.s 157 z-r,!, 156 •2-r,r> 1 55 .'■r>6 154 2-37 153 2:5S 151 2-BU 150 2-60 149 2-61 147 2-(:2 146 2-ßS 145 2-6^ 143 2-65 142 2-r,i] 140 2-67 139 2-6S 138 2-69 136 2-7() 135 2-71 133 2-72 132 2-7.i 130 VX 120 127 2-u; 126 2-77 125 2-7S 123 2-7V 122 2-Sil 120 2-SI 119 2-S2 117 2-S.j 110 2-Si 114 2-SS 113 2-S6 111 2-S7 110 2-S8 108 2-S9 107 2-00 105 2-91 104 2-02 102 2-93 101 2-9J, 99 2-95 98 2-9t! 96 2-97 95 2-98 93 2-99 92 3-00 *(l+n) •9937903 ■993963 1 ■9941323 ■99 12969 ■9944574 ■9940139 •9947664 ■9949151 ■9950000 •99520 1 2 •9953388 •9954729 •9950035 •9957308 ■9958547 ■9959754 ■9960930 ■9962074 ■0963189 ■9964274 ■9965330 ■9906358 ■9967359 ■9968333 ■9969280 ■9970202 •9971099 ■9971972 ■9972821 ■9973640 ■9974449 ■9975229 ■9975988 ■9976726 ■9977443 ■9978140 ■9978818 ■9979476 ■9980116 •9980738 •9981342 •9981929 •9982498 •9983052 ■9983589 ■9984111 ■9984618 ■9985110 ■9985588 ■9986051 •9986501 A + 1731 1688 1046 1005 1505 1525 1487 1449 1412 1376 1341 1300 1272 1239 1207 1176 1145 1115 1085 1056 1028 1001 974 948 922 897 873 849 825 803 781 759 738 717 697 678 658 640 622 604 587 570 553 537 522 507 492 478 464 450 A2 44 13 12 11 40 39 3!) 38 37 30 35 35 34 33 32 32 31 30 2!) 20 28 26 26 24 24 23 23 22 22 21 21 20 20 19 19 18 18 17 17 16 16 16 15 15 14 14 14 13 ■0175283 0170947 (116671)1 01025 15 1)158476 0154193 0150596 Ol 10782 0143051 Ol 39401 0135830 Ol 32337 o|28;)21 01255S1 0122315 0119122 0116001 0112951 01(J9909 O107O50 0104209 Ol Ol 428 0098712 0096058 0093466 0090936 0088405 0086052 ■0083697 ■0081398 0079155 ■0076965 0074829 ■0072744 '0070711 0068728 0066793 0064907 0063007 0061274 0059525 :)057821 0056160 0054541 0052963 0051426 0049929 0048470 0047050 0045666 0044318 4330 4246 4157 40(i9 3982 3897 3814 3731 305(_) 3571 3493 3416 3340 3200 3193 3121 3051 2981 2913 2847 2781 2717 2654 2592 2531 2471 2413 2355 2299 2244 2189 2136 2084 2033 1983 1934 1886 1839 1793 1748 1704 1601 1619 1578 1537 1497 1459 1421 1384 1347 + 92 91 89 88 86 85 84 82 81 80 78 70 74 73 72 70 69 68 67 60 64 63 02 61 60 59 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 40 39 38 37 36 35 Biometrika ii 186 New Tables of tJie Frohnbi/if;/ ftäegral TABLES 1. AND IL— (con38 13 ■0043007 35 S-02 •9987361 13 •(XI41729 34 S-03 ■9987772 12 •0040486 33 S-Oi •9988171 12 ■0U39276 32 S-Oö •9988558 12 •0038098 32 s-on 3-07 •9988933 ■9989297 3(i4 353 342 332 322 11 11 •0036951 ■0035836 1115 1085 1056 1027 999 31 30 s-os S-00 •9989650 •9989992 11 10 •0034751 •0033695 29 29 3-10 •9990324 10 •0032668 28 3-11 •9990G4G 312 302 293 284 275 10 •(«31669 971 944 918 893 868 27 S-12 ■9990957 10 ■0030698 27 3-13 •9991260 9 •1X329754 26 3-li ■9991553 9 •0028835 26 S-15 •9991836 9 •0027943 25 3-16 •9992112 267 258 250 242 235 9 •0027075 843 820 797 774 752 24 3-17 •9992378 8 •0026231 24 s-is 3-10 •9992636 •9992886 8 8 •0025412 •0024615 23 23 SSO ■9993129 8 •0023841 22 3-21 3-22 ■9993363 ■!i:):i3.-i;)0 227 220 213 206 200 7 7 •0023089 ■0022358 731 710 689 669 650 21 21 S-23 ■9993810 7 ■0021649 20 3-2k ■999402-1 7 •0020960 20 S-25 ■9994230 7 •0020290 19 3-26 ■9994429 193 187 181 175 169 6 •0019641 631 612 595 577 560 19 3-27 ■999 1623 6 •0019010 18 3-28 •9994810 6 •0018397 18 S-20 3-30 ■9994991 ■9995166 6 6 •0017803 •0017226 17 17 2-31 •9995335 164 159 153 148 143 6 ■0016666 543 17 3-32 •9995499 5 •0016122 16 3-33 3-34 •9995658 •9995811 5 5 •0015595 •0015084 512 496 481 16 15 335 •9995959 5 •0014587 15 3-36 •9996103 139 134 130 125 121 5 •0014106 467 453 439 426 413 15 3-37 •9996242 5 •0013639 14 3:3S •9996376 4 •0013187 14 3-30 S-J^O •9996505 •9996631 4 4 •0012748 •0012322 13 13 3-4 1 •9996752 117 113 109 106 102 4 •0011910 400 388 376 364 353 13 3-42 •9996869 4 •0011510 12 3-4.3 •9996982 4 ■0011122 12 3-44 •9997091 4 ■0010747 12 3-4.5 •9997197 4 •0010383 11 3-40 •9997299 99 95 92 89 3 •0010030 342 331 320 310 11 5-47 •9997398 3 •0009(589 11 3-43 •9997493 3 •0009358 10 .1-40 •9997585 3 ■0009037 10 1 ,?'.7" ■;i;i:iT(;7i 3 ■0008727 10 X y\+a) A + A» 3-50 •9997674 86 83 80 77 74 72 3 3-51 •9997759 3 3-52 •9997842 3 3-53 •9997922 3 3-54 •9997999 3 3-55 •9998074 3 3-50 •9998146 69 67 65 62 60 3 3-57 •9998215 2 3-5S •9998282 2 3-50 •9998347 ■2 3-60 9998409 2 3-61 9998469 58 56 54 52 50 2 3-6.' •9998527 .) 3-t;.i ■9998583 •2 3-64 •9998637 ■1 3-65 •9998689 2 3-66 •9998739 48 47 45 43 42 2 3-67 •9998787 2 3-6S •9998834 2 .3-69 •9998879 2 3-70 -9998922 2 3-71 •9998964 40 39 37 36 35 3-72 •9999004 1 3-7-J •9999043 1 S-74 •9999080 1 3-75 •9999116 1 3-76 •9999150 33 32 31 30 29 1 3-77 •9999184 1 3-7S •9999216 1 5-7.9 •9999247 1 3-SO •9999277 1 ,;•*'/ •9999305 28 27 26 25 24 1 S-S2 •9999333 I S-S3 •9999359 1 3-S4 •9999385 1 3-S5 ■9999409 1 3-86 •9999433 23 22 21 20 19 1 3-87 •9999456 1 3-SS ■9999478 1 3-89 •9999499 1 3-90 •9999519 1 3-91 •9999539 19 18 17 17 16 1 3-92 •9999557 1 3-93 •9999575 1 3-94 •9999593 1 3-95 •9999609 1 3-96 •9999625 15 15 14 14 1 3-97 •9999641 1 3-9S •9999655 1 .^-99 •9999670 1 4-00 -9999683 1 W. F. SlIKPI'AIlD 187 TABLES I. AND II. — {continued). •0008727 •0008426 •0008135 •0007853 •0007581 •0007317 •00070G1 •00OG814 •0006575 •0006343 •0006119 •0005902 •0005693 •0005490 •0005294 ■0005105 •0004921 ■0004744 •0004573 •0004408 •0004248 •0004093 ■0003944 ■0003800 •0003661 ■0003526 •0003396 •0003271 •0003149 ■0003032 ■0002919 •0002810 ■0002705 ■0002604 •0002506 •0002411 •0002320 •0002232 •0002147 •0002065 •0001987 •0001910 •0001837 •0001766 •0001698 •0001633 •0001569 •0001508 •0001449 •0001393 •0001338 301 291 282 273 264 256 247 239 232 224 217 210 203 196 189 183 177 171 165 100 155 149 144 139 135 130 125 121 117 113 109 105 102 98 95 91 88 85 82 79 76 73 71 68 66 63 61 59 57 55 A2 + 10 10 9 9 9 8 8 8 8 .1' K1+«) A + A2 z A A2 2 4-ii(i ■9909683 13 13 1 »■) 1 ■0(X)1338 53 Jriii ■9099G9(; 1 ■0001286 j,-(h: ■99!»97l)9 ■0001235 Ol 49 47 45 43 "o J,-U.J ■9999721 1 z 12 n 11 ■0001186 2 j,-i)Jt ■9999733 ■0001140 2 Ji-i)5 ■9999744 ■0001094 2 j,->»; ■9999755 10 10 9 9 9 ■0001051 42 40 39 37 36 2 J, 07 ■9999765 ■11001009 2 j,-ns ■9999775 ■000(1969 //■ii'.i ■99:)!I784 ■1)000930 ] .',■10 ■9999793 ■0(.»0U893 4-11 ■9999802 8 8 8 •0000857 35 33 '?0 J,-V'^ ■999981 1 •0000822 1 J,-1S ■999981!» ■0000789 1 //•'■'/ ■9999826 •0000757 31 30 j,-ir> ■9999834 7 •0000726 1 jf-iij ■9999841 ■0000697 28 *>7 4-17 ■9909848 i ■0000668 4- IS ■9999854 7 6 6 6 ■0000641 26 4-19 ■9999861 ■0000615 4-:20 ■9999867 ■0000589 24 4-21 ■9999872 6 ■0000565 23 22 4-22 •9999878 ■0000542 4-2:; ■9999883 r •0000519 '^•^ 4-24 ■9999888 O ■0000498 21 20 4-:r, ■9999893 5 ■0000477 4-2ii ■9999898 4 4 4 4 4 ■0000457 19 18 18 17 16 4-27 ■9999902 ■0000438 4-2S ■9999907 ■0000420 4-20 ■9999911 ■OOOO402 4- so ■9999915 ■0000385 4:n ■9999918 4 3 ■0000309 16 15 14 14 13 4-S2 ■9999922 ■0(X)0354 4' SS ■9999925 ■0000339 4-S4 ■9999929 ■0000324 4-sr, •9999932 3 3 ■0000310 4-SG ■9999935 3 3 3 3 2 ■0000297 13 12 12 11 11 4-S7 ■9999938 ■0000284 ^ 4-ss ■9999941 ■0000272 4-so ■9999943 ■0000261 4-40 ■9999946 ■0(.)00249 h-),i ■9999948 •7 ■0000239 10 10 9 9 9 4-42 ■9999951 z 9 ■0000228 4-4S ■9999953 2 9 ■0000218 ',-4!, ■9999955 •0000209 4-4^ ■9999957 •0000200 4-41' ■|»999959 w) •0000191 8 8 4-47 ■9999961 9 •0000183 4-4S ■9999963 2 •0000175 4-49 ■9999964 •0000167 4-50 ■9999960 •0000160 1 24—2 188 New Tabh's qf flie Prohahility Integral TABLES I. AXU l\.—{continued). See Note, p. 181. .r i,(\■>^a^ z j,-ao G6023 159837 Jt-51 67')8e 152797 J,-52 (i908O M 6(151 Jf-öS 70508 139590 Jrcj, 71873 133401 itT.r, 73177 127473 4-50 74423 121797 j,-->~ 75614 116362 Jrüs 70751 111159 J,-5i) 77838 106177 Jl'OO 78875 101409 4-01 79867 96845 4-G3 80813 92477 ./,-fö 81717 88297 j,-e,j, 82580 84298 4-65 83403 80472 4-06 84190 76812 4-07 84940 73311 4 -CS 85656 69962 4-G9 86340 66760 4-70 86992 63698 4-71 87614 60771 4-72 88208 57972 4-rs 88774 55296 7-74 89314 52739 4-75 89829 50295 476 90320 47960 ^■77 90789 45728 4-7S 91235 43596 4-79 91661 41559 4- flu 92067 39613 4-si 92453 37755 4-82 92822 35980 4-ss 93173 34285 4-S4 93508 32667 4SÖ 93S27 31122 4-86 94131 29647 4-87 94420 28239 4-SS 94696 26895 4-89 94958 25613 4-90 95208 24390 4-91 95446 23222 4-92 95673 22108 4-93 95889 21046 4-94 96094 20033 4-95 96289 19066 4-96 96475 18144 4-97 96652 17265 4-9S 96821 16428 4-99 96981 15629 X i(H-a) z 5-00 97133 14867 5-01 97278 14141 5-02 97416 13450 6-Oä 97548 12791 5-04 97672 12162 5vö 97791 11564 5-06 97904 10994 5-U7 98011 10451 5VS 98113 9934 5-09 98210 9441 5-JU 98302 8972 5-11 98389 8526 5-12 98472 8101 5-13 98551 7696 5-14 98626 7311 5-lü 98698 6944 5-u; 98765 6595 5-17 98830 6263 Ö-IS 98891 5947 5-19 98949 5647 5-20 99004 5361 5-21 99056 5089 5-22 99105 4831 5-23 99152 4585 5-24 99197 4351 5-20 99240 4128 5-2(: 99280 3917 ö'2: 99318 3716 Ö-2S 99354 3525 5-29 993H8 3344 5-3U 99421 3171 5-Sl 99452 3007 5-S2 99481 2852 5-33 99509 2704 5-34 99535 2563 5-3Ö 99560 2430 5-S6 99584 2303 5-37 99606 2183 5-3S 99628 2069 5-S9 99648 1960 5-40 99667 1857 5-41 99685 1760 S-42 99702 1667 6-J^ 99718 1579 6-44 99734 1495 B-45 9974S 1416 5-40 99762 1341 5-47 99775 1270 5-48 99787 1202 5-49 99799 1138 X *(!+'") z r.-üo 99810 \Qrn 0-51 99821 1019 5-52 99831 965 5-r,s 99840 913 5-54 99849 864 5-55 99857 817 5-M 99865 773 5-Ö7 99873 731 5-öS 99880 691 5-Ö9 99886 654 OVO 99893 618 5-61 99899 585 5-62 99905 553 5-G3 99910 522 r,-64 99915 494 ö-t!ö 99920 467 5-66 99924 441 ö'67 99929 417 5-68 99933 394 5-G9 99936 372 5-70 99940 351 5-71 99944 332 5-72 99947 313 5-73 99950 296 Ö-74 99953 280 5-7Ö 99955 264 5-76 99958 249 5-77 99960 235 5-78 99963 222 5-79 99965 210 5-80 99967 198 5-81 99969 187 5-82 99971 176 5-S3 99972 166 5-84 99974 157 5-Sö 99975 148 5-SG 99977 139 5-87 99978 131 5-SS 99979 124 5-S9 99981 117 5-90 99982 110 5-91 99983 104 5-93 99984 98 5-93 99985 92 5-94 99986 87 5-95 99987 82 S-96 99987 77 5-97 99988 73 '■!IS 99989 68 't'9',* 99990 65 6-00 99990 61 W. F. SlIKI'l'ARD .189 TABLES III. ANU IV. Äbsvissa und Ordinate in terms of difference of Areas. ■00 ■Ol ■0-2 ■OS ■0.1, ■0') ■Od ■07 ■OS ■OH ■10 ■11 ■! ; ■1-1 ■i-', ■tr, ■n; •77 •;.s' ■10 ■20 ■ji ■26 ■27 ■as ■29 ■SO ■31 ■S2 ■SS ■SJt ■35 ■SO ■S8 ■S9 ■40 ■4~' ■43 ■h-h ■J,5 ■1,0 ■J,7 ■J,8 ■49 ■50 •0000000 •0120335 •0250G89 •037{)083 ■0.'.01.'-)3(i ■O(;27068 ■0752099 ■0878448 ■1004337 ■1130385 •1256613 •1383042 •1509692 ■1 036585 ■1763742 ■1891184 •2018935 •2147016 •2275450 •2404260 •2533471 ■2663106 ■2793190 •2923749 •3054808 •3186394 •3318533 •3451255 •3584588 •3718561 ■3853205 •3988551 •4124631 •4261480 •4399132 •4537022 •4676988 •4817268 •4958503 ■5100735 ■5244005 ■5388300 ■5533847 ■,-.i;s(j515 ■5S2S415 ■5977601 ■6128130 ■6280060 •0433454 ■0588377 ■11744898 A + 125335 125354 125394 125453 125532 125631 125750 125889 126048 120228 1 20429 120050 120893 127157 127443 127751 128081 128434 128811 129211 129635 130084 130559 131059 131580 132140 132722 133333 133973 134044 135340 130081 136849 137052 138490 139300 140281 1412.35 142231 143271 144355 145487 140068 147900 149180 150529 151930 153394 154923 156521 A2 + 20 39 59 79 99 119 139 159 180 201 221 2 13 204 286 308 33ij 353 376 400 424 449 474 500 527 554 .582 611 040 671 702 735 708 803 839 870 914 954 996 1039 1085 1132 1181 1232 1286 1342 1402 1404 1529 1598 1670 A3 + 20 20 20 20 20 20 20 20 20 21 21 21 21 22 22 22 23 23 24 24 :;5 20 28 29 30 30 31 32 34 35 ;!() 37 39 40 12 43 45 47 49 51 54 56 59 62 05 09 72 •3989123 •3989109 •.3988169 •3980003 •3984408 •3981.587 ■.■5978138 ■3974000 •3909353 ■3904016 ■3958049 •3951450 •3944218 •3936352 ■3927852 •3918715 •3908939 •3898525 •3887469 •3875709 •3803425 •3850434 •3830794 ■3822501 ■3807555 ■3791952 ■3775090 ■3758706 ■3741177 ■3722919 ■3703990 ■3084380 ■3604103 ■3043138 ■302 1487 •3599140 •3570109 •3552374 •3527935 •3502788 •3470926 •3450346 •3423041 •3395005 •3306233 •3330719 •3306455 •3275435 •324.3052 •3211098 •3177766 313 940 1507 2194 2821 3449 4078 4707 5337 5967 6599 7232 7860 8501 9137 9775 10415 11056 11099 12344 12991 13641 14292 14940 15003 16262 16924 17.i89 182.58 18929 19004 20283 20905 21651 22342 23036 23735 24439 25148 25861 26580 27305 28035 28772 29514 30204 31020 31783 32554 33333 027 027 027 027 627 628 628 029 (i30 631 632 633 034 635 630 <)38 640 041 643 045 047 049 052 054 657 659 662 665 668 072 675 679 082 080 (i90 095 699 704 709 714 719 725 730 736 743 749 756 763 771 779 787 A3 lyo New Tahles of thc Prohnbilitii Integral TABLES III. AND lY.—icontinued). ■50 ■51 •ÖS ■53 ■5k ■56 ■57 ■58 ■59 ■m ■61 ■üä ■63 ■64 ■65 ■66 ■67 ■68 ■69 ■70 n ■72 ■73 ■76 ■77 •78 79 ■80 •G7 14898 ■(;iK)3088 ■7UÜ3026 •7224791 ■7388468 •7554150 •7721932 ■7891917 •SIW4212 •8238936 •841G212 •8596174 •8778963 •8964734 •9153651 •9345893 •9541653 •9711139 •9944579 1 -Ol 5-2220 1 036 1334 10581216 1 0803193 1 •1030626 M-2639U 11 503494 M7498C8 1 2003589 r2265281 1 2535654 12815516 + 158191 159937 161765 163678 165682 167782 169984 172-296 174724 177276 179961 182789 185771 188917 192242 195760 199486 203440 207641 212114 21688-2 221977 227432 233286 239583 246374 253721 261693 270373 279861 + 1670 1747 1828 1913 2004 2100 2203 2312 24-28 2552 2685 2828 2981 3147 3325 3518 3727 3954 4-201 4472 4769 5095 5455 5854 6297 6792 7347 7972 8681 9488 10414 A3 + 76 81 86 91 96 102 109 116 124 133 143 153 165 178 193 209 227 248 271 297 3-26 360 399 443 495 555 625 709 8(18 926 3177766 3143646 3108732 3073013 3036481 2999125 2960936 2921902 2882013 2841256 2799619 2757089 2713653 2669295 2624000 2577753 2530535 2482330 2433117 2382877 2331588 2279226 2225767 2171185 2115451 2058535 2000405 1941024 1S80356 1818357 1754983 34119 34915 35719 36532 37356 38189 39034 39889 40757 41637 42530 43437 44358 45295 46247 47217 48-205 49213 50240 51289 52362 53459 54582 55734 56916 58130 59380 60669 61999 63374 787 795 804 814 823 834 844 856 867 880 893 907 921 937 953 970 988 1007 1028 1049 1072 1097 1123 1152 1182 1215 1250 1288 1330 1375 14-25 A' 9 9 9 10 10 II 11 12 12 13 14 15 15 16 17 18 19 20 22 23 25 26 28 30 33 35 38 42 45 50 VAKIATION IN "EUPAGURUS (HELLEK). PRIDE AUXI" By E. H. SCHUSTER, B.A. I. Introduction. The work on which thc tollowing paper was bascd was donc in the wintcr of lüOl — 1902 iliuiiig my occupancy of the Oxford Univcrsity table at thc Naplcs Biological Station ; and I takc this opportunity of exprcssing my appreciation of the kindness and courtesy of tliose members of the .stafl" of that institution with whom I canic into contact. The subsequcnt calculations werc done iindi-r the dircction of Professor Weldon, and I herewith tender hini my sincerc thanks for devoting so miuh timc and trouble to the purpose. I have also to thank Professor Pearson für bis uumerous and valuable suggestions. Thc work itself is an attempt to determine whether members of thc species Ewpagurus pndeauxi caught in shallow watcr, differ with regai-d to certain charac- ters from those caught in comparatively dcep water. For this purpose the following three measureinents were taken : 1. From the upper articulation of the propodite with the carpopodite of the right chehi to the upper and outer articulation of thc propodite with the dactylo- podite, called Measurement No. 1. Fig. 1, AB. -C A-- Fio. 1. J!I2 Variation in " Btipagums Prideaiixi'" 2. The length of the right chela measured t'rom its lower articulation with the carpopodite, to the furthest point of the fixed blade of the scissors. Fig. 1, CD. 3. The length of the carapace along the median line. These nicasurements werc takeii with a pair of dividors provided with a screw tine adjustment and an ivory scale diviiled into ^ niillinietres, they are believed to be aceurate to -^ of a miliimetre. From Me;isurements No. 1 and No. 2 an index called ihe chela index was dcdnced, which is intended to represent the proportion wliich tlie claw bcars to the wholc length of the chela. The chela index = Meiisurement No. 2 — Measurement No. 1 Measuiement No. 2 About two thousand individuals were measured, which were separated into two main groups : I. Shalloiu water forms, fiom a depth of 35 metres or under; II. Deep water /onus, from a depth of over 85 metres. Owing to the great ditference in size between the sexes each of these groups was again subdivided into male and femalc. Thus finally we have to deal with four sets with about fivc hundred individuals in each. IJ. Compariiiun between deep water forins and shallow water furnis an regards each measurement and the chela index taken separatehj. The deep water males are in each case compared with llie shallow water males and the deep water feraales with the shallow water fcmales. TABLK I. Measurement No. 1. (/'. Mean Probable error of | Standard Deviation raean Deep water forms ... Shallow water forms 9-7078 mm. 10-2718 mm. -0847 mm. 27588 mm. -0745 mm. 2-5901 mm. Diflerence 0-5640 mm. Probable error of diflerence = -1 1 28 mm. This measurement is greater in the shallow water forms by an amount exactly five times as great as its probable error, and therefore almost certainiy significant. E. H. J. Schuster 193 TABLE II. Measitreinent No. 1. $. Mean ProlialiU' error of mean Standard Deviation Deej) \v;itor forms . . . Shallow watcr forms 7-4000 mm. 7 •48.^)0 mm. ■O.-J.-JO mm. -02'J3 mm. 1-0607 mm. 1 0:234 mm. Difterence 0-0850 üini. Probable error of differeiice = -0441 mm. This measurement is greater in the shallow water forms by an amount about twice as great as its j)n)bable error. TABLE III. Itiffht Chela Lemjth. ^. ,, Probable error of ^ mean Standard Deviation Deep watcr forms ... Shallow water forms 17-9676 mm. 18-6773 mm. -14.51 mm. ■1259 mm. 4-7279 mm. 4-3769 mm. Difference -7097 mm. Probalile error of difference = -1922 mm. This measurement is greater in the shallow water forms by an amount between three and four times as great as its probable error, and therefore is probably significant. TABLE IV. Rigid Chela Length. ?. , Mean Probable error of „. j j t\ • i- mean Standard Deviation Deep water forms ... | 14-1415 mm. Shallow water forms \ 13-9735 mm. -0612 mm. -0522 mm. 1-9679 mm. 1-8215 mm. Difference -1680 mm. Probable error of difference = -0805 mm. This measurement is greater in the deep water forms by an amount twice as great as its probable error. To sum up with regard to the ab.solute measurements of the right chela; in the male they are both significantly greater in the shallow water than in the deep water forms ; while in the female Measurement No. 1 is greater in the shallow water forms, Measurement No. 2 in the deep water forms, and in neither case is the difference, thougli well marked, sufficiently great to be calied significant. Biometrika ii 25 194 Variation in " EnpaginniA Prideauxi" TABLE V. Chela Index. ^. Mean Probable error of mean Standard Deviation Deep watcr forms . . . Shallow water foniis •461348 ■451045 ■000538 •000472 •017537 •016417 Difference •010303 Probable error of difFei'ence = •000715 The chela index is greater in the deep water forms by an amount more than 13 times as big as its probable error, antl tberefore certainly significant. TABLE VI. Chela Index. J. Mean Probable error of mean Standard Deviation Deep water forms ... Sha low water forms ■475051 •471312 •000458 •000385 ■014721 •013429 Difference •003739 Probable error of difference = ■000598 The chela index is greater in the deep water forms by an amount more than six times as big as its probable error, and therefore almost certainly significant. Thiis Tables V. and VI. show that in the forms taken from deep water, both male and fcmale, a greater proportion of the wholo length of the chela is taken iip by the blade of the scissors thau in forms taken from shallow water. TABLE VII. Carapace Length. ^. Mean Probable error of mean Standard Deviation Deep water forms ... Shallow water forms 8-5854 mm. 8-4063 mm. ■0512 mm. •0425 mm. 1^6696 mm. 1-4918 mm. Difference •1791 mm. Probable error of difference = •0665 mm. The mean carapace length in the deep water forms is greater than that of the shallow water forms by an amount mnrp than twicc as great as its probable error. E. II. J. SCHUSTPm TABLE VIII. Carapace Length. ?. 195 Mean Prubiilile error of meati Stiindiird Deviation Deep water fonas... Shallow water forma 7"5405 mm. 7-1222 mm. -0293 mm. -0247 mm. -9417 mm. -8631 mm. Difl'ereuce -41(H3 mm. Probable error of difference= 0383 mm. The mean carapace length in the deep water forms is greater than that of the shallow water forms by an amount about ten times as great as its probable error. Thus in both male and female the mean carapace length is greater in the deep water forms, in the female by a certainly siguificant amount, in the male by a qiiantity which taken alone could hardly be taken as significant, but viewed in the light of the result obtained by the female it may possibly be considered to be so. III. On the Comparative Variuhility of Deep und ShaUoiu Water Forms. TABLE VIII bis. Table Deep Water Shallow Water Difference of Character, Sex S. D. C. of V. S. D. C. of V. S. D.'s C.'s of V. Measiirement No. 1 ^ \ 2-7588 + -0599 1 -0607 ± -0233 28-418 + -665 14-334 ± -322 2-5901 + -0527 10234 ± -0207 25-216 + -544 13-673 ±•278 •1687 + -0798 -0373 ± -0312 3-202 + -865 •661 ±-425 Right Chcla Length 1-6696 + -0362 -9417 ± -0207 19-446 ± -438 12-489 ±-279 1-4918 + -0303 -8631 ±-0175 17-746 + -372 12-118 ±-249 •1778 + -0470 •0786 ±-0271 1-700 + -575 •371 ± -374 Chela Index ^t j 9 * -01754 ±■00038 -01472 ± -00032 38-013 + -936 30-988 + -744 -01642 + -00033 -01343 + -00027 36-404 + -833 28-493 + -622 •00112 ±-00051 -00129 ±-00042 1-608 + 1-2.53 2-495 ± -970 25—2 10(5 Variation in "Eupagurva Prideanxi" The table above gives the Standard deviations and the coetficients of Variation* with thcir probable crrors, and we see : (i) That in buth f'oruis the male, whether we judge by Standard deviation or coefficieut of Variation, is, for the characters considered, niuch inore variable than the female. (ü) That there are significant differences in the variability of the deep and shallow water forins. The deep water fortns are iu evcry single case the niore variable, however variability be estiniated. For males the difiference is always greater and often miich greater than its probable error ; for females the differeuce is iess marked, but none the less quite obvious. We can therefore conclude ihat the couditions of life are probably far niore stringent for the shallow than the deep sea forms, and for the females than for the males. It would, perhaps, be rash to assert that the shore crabs are a selection from the deep sea form, but the facts as to variability are not ouly compatible with but indeed suggestive of such an hypothesis. IV. Gomparison of the correlation between the length of carapace and euch uf the other measuremeiits for deep und shallow tuater forins. To commence with the consideralion of the males, Table IX. shows the correla- tion between the length of the carapace and Measurement No. 1 for deep. Table X. for shallow water forms. For the former r = -G+.iS + ■0032, For the latter r = "9337 + 0037, The difference = Ol iO ± 0049. * Percentage Variation on the luean, i.e. 100 x Standard deviation and dividcd by the mean. E. H. J. Schuster 107 X ^ El. w *-; l-J _ o < "^ H ■^ a .JO Ü o t o "4 (1) ?- ^ O 'Ss f^ Totals fM .C OC' M t- C1 — ^^ — ■ t^ 05 o cc cc o « ^ ^ :c Ol ic 'C -t 1.-; r3 -i" -r Ol ^i ^ 9-ll-r-ll - 1- 1 1 1 1 1 1 1 1 1 1 1 1 1 1 '- 1 1 M 1 1 1 1 1 1 1 1 1 1 fM 0:U".>-UI - S-91-1-91 1 cc I i-H 1 1 { 1 1 1 ! 1 1 1 1 1 -t 0-91-9-91 -" 1 1^ 1 1 1 1 M 1 1 1 1 1 in 9-9l-T-in 1 ?0 (M 1 { 1 I 1 1 1 1 1 1 1 1 1 9-fl^I-ff l^lO-tr-llllllllllll 1 { CD CO 1 1 1 1 1 { 1 1 1 1 I 1 Ol O-n-9-Sl l^cocDoqlllllllllll 5^ s-si-i-er iii'~'^^llli|]llii 0-fT-9-l!r ] ji— »cMcoG^col 1 1 [ 1 1 ; 1 [ 9-^l-l-cl lll.-.i>^co|llilllll o-ci-n-n 1 1 1 .-H I-- 00 rH 1 1 1 1 1 1 1 1 1 9-ll'l-ll [lil^aDin-— 1|||| llll Ol (l-Tl-9-fll 1 1 1 1 1 n-( rH 1 1 1 1 1 1 1 1 9-Ol-l-Ul llll,— iicr^iOr-il II III ! llll i-H 1— < 1 1 1 1 1 1 1 0-01-9-G 1 1 1 1 1 r-H G-l t^ CD 1 1 1 1 1 1 1 CD Ol 9-6 -1-6 ||||||CNi— 'tOC0|| llll n 0-6 -9-8 1 1 Iwr^l [(MOC-^^I 1 1 1 1 OJ 9 3 -T-Ü [ 1 1 1 1 1 1 fM t^ >— 1 O 1 1 1 1 1 1 1 1 1 1 1 1 (M ^ 1 1 1 1 1 5 n-s -9-1 1 1 1 1 1 |rM|r-iCDOC0| 1 1 1 1 1 1 1 1 1 I (N llll oq 9-1 -I-l 1 1 1 1 1 1 1 |i— ii-t-r}*eo.-i| 1 1 1 1 1 1 1 1 1 1 <-• (N III o O-l -9-9 1 1 1 1 1 [r-«|r-4|O^OI>| 1 1 Ol 9-9 -1-9 1 1 1 1 1 1 1 1 II 1'':^=° 1 1 Ol O) Ol O) 0-9 -9-9 1 I 1 1 1 1 1 1 rH 1 tM rf r^ Ol « 1 9-9 - r-9 1 1 1 1 1 1 1 1 1 1>1 1 1 ^ tM O rH 0-9 -9-f 1 M 1 II 1 1 II M 1 1^- n 9-f -T-f 1 1 II II M ! M II- 1- (M '-T ::: »-■:) o ^^-t C: >o o >-■: c: ic ^ »-■: c; i:: c^ 1 e2 '^j f^^ "^ *-i o ^. c: Ci X' ^ e- i-^ "o "o »--^ »--^ 7"7777 M M M M 1 1 '^ Co *-H cc *^ ^ »^ "p- '^ 'p »7^ 'p '^ '-c "7^ "P (^^i '^ T^ Ö i^ C; ci öc io i-- t- "ö i VC lo -^ •(s8j:j8uii]juu) ao^dureQ jo q'}Su8''j 198 Variation in " Eupaguriis Pticleauxi" ^ s 8 ^ S =^ Ol t3rtFH| 1 1 1 1 1 1 1 00 9-cl-lcT 1 1 'M -M — r; -ji 1 1 1 1 1 1 1 1 1 (J-ol'9-ll 1 1 l-^S'- ! M 1 1 1 1 1 1 9-ii-i-n 0-IJ-9-0T 1 1 ^ l"22 1 1 1 1 1 1 1 1 1 1 |||.-CCC3CSr-c||||||| s? 9-01-1-01 1 1 1 1 1 ri »< 1 1 1 1 1 1 1 1 00 0-01-9-0 ll|i|(N>nt^O||||||| 1 1 1 1 1 ^ —I 1 1 1 1 1 1 1 -1* o 9-6 -T-6 II ||ir-.(NX o s ■4-3 1) •s^ s ö:; li ^ ü < -4^ bf) H g •« P5 -o o o -C Ö bC O <» o h hJ o !. ^ ^ o Ol o QO ^5 1~ (>) r- 03 — 1 1^ 33 O M M O TO — ^eo'^^lOo-*lOc«^t^(^^!^^r- PO oo t 6-oy-iuiK " 1 '^ 1 1 M 1 1 1 1 1 1 1 1 1 o» 6-6o-0-Gö M 1 1 1 1 1 M 1 1 1 1 M 1 1 6-8o-0-8S 1 -° 1 1 M 1 1 1 M 1 1 1 M lO 6-lS-0-l~ ^ lO Ol « 1 1 1 1 1 1 1 1 1 1 1 1 6-9S-0-9o |mtO'*.-'|||||||li|| r—t 6-SS-O-So |(Nio^m||||||||||| ^ e-fs-o-fs ||«VOC>0.-H|||||||||| 64-fy-O-Po 1 [ .— 1 CO >C Q-l j j I [ [ I 1 1 1 1 - 6-S-S~0-^(j |||^CO<33Co|||||]||| Ol 6-U-O-lc 1 1 1 (M CD CD K ^ 1 1 1 1 1 I 11 OO Ol 6-Oe-o-or ||||.-HQO.-i-Hl||l|||| G-Cl-D-Gl 1 1 1 1 -- --^ S '--^ ^ M M 1 M 6-Sl-O-Sl 1 1 1 1 1 "^^5" 1 1 1 1 1 M 6-U-O-lI 1 1 1 .-H 1 1 O) CD O 1 1 1 1 1 1 1 III II ^ G-1 1 1 ! 1 1 1 1 ^ 6-0I-0-0I 1 [ 1 [ 1 [ 1 ^ O tt>l CM 1 1 1 1 1 6-9l-0-'JT 1 1 1 l'-'l |i-l^t~T)i| 1 1 1 1 6-fl-O-fl 1 1 1 1 1 1 (M 1 1 O M t- 1 1 1 1 1 1 1 1 1 1 II (M 1 1 1 1 1^ n 6-9T-0-n 1 1 1 1 1 ir-i|oirtooiom| 1 1 o 6-^T-O-^T |||||i||{||oca3(M|| OS 6-ii-n-ii 1 1 1 1 1 1 1 1 rH 1 (N 1 03 « 1 1 Ol 6-oi-n-or ! 1 1 1 1 1 1 M- M-t-^ 1 00 6-6 -0-6 111111111-11-1-*^ 6-S -0-8 1 1 M 1 1 1 1 1 M 1 1 - 1- Ol i;^. c> ip ^, ip 'i^ ip c^, u-; CN u^, .::^ L:; c^ '-'t' c^ t^i "^i ■^ »--1 i Öl ci ci öo ^ i'- i^ ^ö 'ö ib ^ 111111 1 1 M 1 M 1 M ^ »^ ^ <-( »-H e2 •(saj'jaiuiuiui) ao^dui^Q jo qijSua'j 200 Variation in "Enpagums Prideauxi" X <1 -T^ e r^ « o m ■fci p O "p Cl i;1 O >--3 9s lO i fl 9 1.-3 TTTT T M M M M M 1 -, -< -1 ~< 1 •(saj'janniiun) aoijiliuu, > jo q'jSna'j E. H. J. Schuster 201 Diagrain I. sIhjws thc regressioii lincs of Meusuromcnt No. 1 (in tlic carapace for thesc twü dopths ; tho ilotttMl liiio is tor tlie deup water foniis, the ooiitiiiuoiis lino for tlie .shallow watcr forins. It will bt; iiDÜced that one carapace Iciigth is as.sociated wiih a iiiucli smallcr value of Measuruiiicut No. l in the dcep water fonii.s thaii in tili' .shallow. Di.UiRAM I. ^[l^usu)■^•nu•llt Xo. I. mm 6 3 6 3 7 3 8 3 9 3 10 3 11 3 12 3 13.3 14 3 16 3 16 3 12.3 11.3 10.3 ^ ^ ^ ^■y^ B D B y' y X -^ y® "/' / , ^- D n D p. 1 a X e.3 6.3 r D B •;? X <^ Comparison between Regression Lines of Me;isui-ement No. I on Carapace Lengtli between Shallow Water