242 



Prof. Karl Pearson. 



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 reversion in Basset 

 Hounds, Mr. Bramley-Moore indicated to me how my method, although 

 correct in theory, differed sensibly in the numerical results with the pro- 

 cesses of interpolating employed. 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. 

 Thus the thirty-six coefficients of heredity for coat-colour in horses and 

 eye-colour in man have been re-calculated, as well as twelve coefficients 

 for heredity in coat-colour of Basset Hounds given in a paper on the 

 Law of Reversion presented on December 28th, and about to appear 

 in the ' Proceedings.' The great growth of the theoretical investigations 

 has, however, compelled me to break up the old memoir* of last August 

 into two parts, the one (the present) dealing only with theory, and the 

 other with its application to inheritance in the horse and man. 



2. The theory of the present memoir depends upon a very simple 

 feature of normal correlation. If z8xi8x2 .... Sxn be the frequency 

 of a complex of characters lying between Xi and xi + Sxi, X2 and 

 X2 + Sx2 . . . Xn and Xn + 8xn, whcrc Xp is the deviation of the ^th 

 character from its mean, then 



dz dh 



dVjpq dXp dXq 



where Tpq is the correlation of the pth and qth organs. 



This simple differential relation enables us to expand z for any 

 number of characters in powers of the correlation coefficients (neces- 

 sarily 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 corresponding organs, the coeffi- 

 cients 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 7i-uncorrelated organs, 

 converts it into a correlated surface of frequency with ^n{n-l) 

 coefficients of correlation of arbitrary values. This operator gives us by 

 aid of certain symbolic equations the expansion of the ?i-fold integral 



!oo 1 00 Too /«ao 

 I zdxidx2dxs dxn 



hi Jh.2 Jh^ jhn 



* That memoir was at my own request returned for revision after being 

 accepted for the ' Philosophical Transactions.' 



