MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 289 



2y, and Sy, can be found in the usual manner by squaring after the insertion of 

 the numerical values. 



From (Ixxxi.)-(lxxiii.) the probable errors and correlations of the moments can be 

 found if required, the calculation being numerically somewhat laborious, but presenting 

 nothing of novelty. 



The probable error of the criterion 



K = 3ft - 2/3, + 6, 



where ft = pi/fd and ft = /u 4 //4 may be found as follows : Put c = (w, + 1) (in, + I), 

 r = nil + m -i, + 2 ; then we find 



and accordingly 



A / 2 2 .1 1 \ / 1 



/ + 1 + e e 



r + I 



A 



r + 2 ,- + 8r + l + (/+! + e) (/ t + I) 



./A __ 2 _ _1 J_ J' + L 



V / ' " r + 2 " / '+ 3 1r r + I' + e ~~ (/ + 1 + e) (m, + I} L 



= ii Am, + i> Aw a ............... (cxx.), 



where z\ and t : admit of easy calculation. Hence 



S*/K 2 = tiS;,, + ^S; la + 2MoS, 1 , 1 S M ,R (lll , 11J ..... (cxxi.). 



The value of S can thus be found, and the steadiness of the curve to its type 

 ascertained. 



Illustration. Glands of the Fore-legs of Swine. 



In the ' Proceedings of the American Academy of Arts and Sciences,' vol. 32, 

 p. 87, 1896, is a memoir by C. B. DAVENPORT and C. BULLARD, on the variation in 

 number of the Mtillerian glands in the fore-legs of 4,000 swine. The paper especially 

 attracted our attention, because the authors are content to describe the frequency 

 distribution of these glands by means of a normal curve. They write, after 

 discussing the plotting of the normal curve on their diagram (pp. 90-91) : 



"These and other characters of the ' probability' curve are indicated in that shown 

 in dotted line* in the accompanying diagram. The diagram also shows the curve of 



* The authors actually represent the normal curve by an 18-sided polygon. 

 VOL. CXCI. A. 2 P 



