March 22, 1900] 



NATURE 



505 



doubting the usual explanation of this effect, which refers it to 

 interaction with the residual impurities of the solvent, the 

 phenomena being different to those observed in the case of 

 permanganate, a salt showing a somewhat similar drop in the 

 curve, for which the usual explanation is satisfactory. Potassium 

 bichromate gives a curve consisting of two well-marked parts — 

 perhaps owing to a change in the nature of the ions at different 

 concentrations. Mr. Griffiths is engaged in measuring the 

 depression of the freezing-point of corresponding solutions, and, 

 when his results are published, a comparison of the two sets of 

 values will be made. 



March i. — "Mathematical Contributions to the Theory of 

 Evolution. VIII. On the Correlation of Characters not 

 Quantitatively Measurable." From the Department of Applied 

 Mathematics, University College, London. Presented by Karl 

 Pearson, F.R.S. 



In August last I presented to the Royal Society a memoir on 

 the inheritance of coat-colour in thoroughbred horses, and of 

 eye-colour in man. This memoir, which was read in November 

 of last year, presented the novel feature of determining correla- 

 tion between characters which were not capable a priori of 

 being quantitatively measured. The theoretical part of that 

 memoir was somewhat brief, but I showed by illustrations that 

 the method could be extended to deal with problems like the 

 effectiveness of vaccination and of the antitoxin treatment in 

 diphtheria. More recently, in studying the phenomena of re- 

 version in Basset Hounds, Mr. Bramley-Moore indicated to me 

 how my method, although correct in theory, differed sensibly in 

 the numerical results with the processes of interpolating em- 

 ployed. I then proposed a new method, and the analytical 

 discussion of its details was worked out in part by Mr. Bramley- 

 Moore himself, by Mr. L. N. G. Filon, M.A., and by myself. 

 Dr. Alice Lee also came to our assistance, and the result is the 

 present joint paper. On the basis of the new methods, we have 

 already worked out upwards of sixty coefficients of correlation, 

 principally of heredity. 



The theory of the present memoir depends upon a very simple 

 feature of normal correlation. If z^x^x,^ .... 5x„ be the fre- 

 quency of a complex of characters lying between x^ and x^ -f Ix^, 

 Xo and Xn -I- Sx^ . . . . , jr„ and Xn 4- 5x,„ where Xp is the deviation 

 of the /t'h character from its mean, then 

 dz ^ cPz 

 drpg dxpdxq 

 where rp, is the correlation of the /th and ^th organs. 



This simple differential relation enables us to expand z for 

 any number of characters in powers of the correlation coefificients 

 (necessarily less than unity) by Maclaurin's theorem. But since 

 we may replace a differential with regard to a coefficient of 

 correlation by a double differential with regard to the corre- 

 sponding organs, the coefficients of correlation may be put 

 zero before instead of after the differentiation. In other words, 

 we obtain a symbolic operator which, applied to a normal 

 surface of frequency for «-uncorrelated organs, converts it into 

 a correlated surface of frequency with A«(«- l) coefficients of 

 correlation of arbitrary values. This operator gives us by aid 

 of certain symbolic equations the expansion of the w-fold 

 integral 



J^J J\ i*« 



■.dx^dx^dx^ dx„ 



in terms of the ^n{n - i) coefficients of correlation, and a series 

 of new functions which we term the ^-functions. These satisfy 

 the difference equation : 



Vn = XVn-i - (« - I)z'h-2 



and the differential equation 



dv„ 



dx 

 The calculation of these functions is shown to be easy, and 

 their properties are investigated. In this manner the volume 

 of a frequency surface of the wth order cut off by n planes 

 parallel to the n co-ordinate planes is shown to be capable of 

 calculation, and its value is determined in the numerical illustra- 

 tions given for example of l, 2, 3 up to 6-fold correlation. It 



may be noted that by putting h^ — h^ = h^=: =^„ = o, we 



have really obtained a result which enables us to find the "area " 

 of a "spherical triangle" in «-fold hyperspace in terms of a 

 series ascending by powers and products of the cosines of the 

 angles between its faces. 



The application of these results to the correlation of characters^ 

 not quantitatively measurable, arises from the fact that the 

 «-fold integral above given, and which we have shown how to- 

 evaluate, measures the total frequency beyond certain boundaries. 

 We can observe, for example, whether horses' coats are bay and* 

 darker (or chestnut and lighter), whether eyes are grey and lighter 

 (or, dark grey and darker). Thus by forming mass frequencies 

 instead of frequency distributions for small changes of character, 

 we can find equations to determine the correlation. The prob- 

 able error of such correlation, the convergency of the series, and 

 other points are investigated. 



A iiumber of illustrations of the new method are given from 

 heredity in horses, dogs and man, and it is shown how normality 

 of frequency must even for such a character as stature ^ only be 

 looked upon as a first approximation. 



An investigation is also made into the influence of superior 

 stock in producing superior offspring. It is shown, for 

 example, that if an individual who possesses a degree of 

 character only found in one in twenty be considered "ex- 

 ceptional," then eighteen times as many exceptional men wilP 

 be born of non-exceptional parents as of exceptional parents ;; 

 but, on the other hand, exceptional parents produce exceptional 

 offspring at a rate ten times as great as non-exceptional parents,., 

 the greater gross product of the latter being due to tlieir much- 

 greater numbers. In other words, distinguished parents are 

 more likely to have distinguished offspring than undistinguished* 

 — ten times as likely— and yet only one distinguished man in nine- 

 teen will be born of distinguished parents. The importance of 

 such conceptions for both natural and artificial breeding cam 

 hardly be over-estimated. 



Chemical Society, March i.— Prof. Thorpe, President, 

 in the chair. — The following papers were read : — On pila- 

 carpine and the alkaloids of jaborandi leaves, by H. A. D^ 

 Jowett. The jaborine of commerce is shown to be a mixture 

 of isopilocarpine, pilocarpidine and a trace of pilocarpine wikb 

 colouring matter ; no evidence has been obtaii>ed of the exist- 

 ence of the alkaloid previously described as jaborine. — Isomeric 

 partially racemic salts containing quinquevalent nitrogen^ - 

 Hydrindamine bromocamphorsulphonates, chlorocamphorsu'r^- 

 phonates and cis 7r-camphanates, by F. S. Kipping. The author 

 explains the formation of two isomeric salts of externally com- 

 pensated hydrindamine with bromocamphorsulphonic, chloio- 

 camphorsulphonic or m->r-camphanicacid as due to partial race- 

 mism.— New syntheses of indene, by F. S. Kipping and H. Hall. 

 Hydrindamine hydrochloride decomposes almost quantitatively, 

 into indene and ammonium chloride when heated. — Potassium ^ 

 nitritohydroximidosulphates and the non-existence of dihydroxy- 

 lamine derivatives, by E. Divers and T. Haga. Raschig's ; 

 dihydroxyiamine derivatives are merely crystalline compounds 

 of potassium nitrite with the 2/3 and 5/6 normal potassiunv, 

 hydroximidosulphates. — Identification and constitution of 

 Fremy's sulphazotised salts of potassium, by E. Divers and T. 

 Haga. — Some acids obtained from a-dibromocamphor, by A.. 

 Lap worth and E. M. Chapman. Camphonic acid, CtoHjsOj^ 

 obtained by hydrolysing a-monobromocampholid, is a ketonic 

 acid yielding an oxime, a semicarbazone and phenylhydrazones ;: 

 it may be converted into tribromocamphonolactone, which facl 

 indicates that bromocamphorenic acid has the constitution. 



/CHa.CMeax 

 Cni >CMe.COOH. 



■\ 



CBr 



CH/ 



— Spectrographic studies in tautomerism. The absorption, 

 curves of the ethyl esters of dibenzoylsuccinic acid, by W. N. 

 Hartley and J. J. Dobbie.— The curves of molecular vibrations 

 of benzantialdoxime and benzsynaldoxime, by W. N. Hartley 

 and J. J. Dobbie. — On campholytic and isolauronolic acids, by 

 J. Walker and W. Cormack. As a result of further work, the 

 authors consider that campholytic and isolauronolic acids are 

 stereoisomeric, their structure being best represented by the: 

 formula 



C(C02H):CMe^ 



I 

 CH, 



CHj 



CMcj. 



NO. 1586, VOL. 61] 



— The configuration of the camphoric acids, by J. Walker ancif' 

 J. K. Wood. — The constitution of camphoric acid, by T- 

 Walker. The author considers the Perkin-Bouveault formula 

 for camphoric acid to be the most satisfactory hitherto proposed., 

 1 Cited by so many as an example of " normality. ' 



