MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 233 



Secondly, 



= III ^T dx l dx, . . dx M , 



d ( ,, 

 dc. g 



= dnjdc a , 



where n is the total number of individuals measured, which is independent of c,. 

 Therefore D s = 0. 

 Thirdly, 



This will not as a rule vanish. If the frequency be normal, it will still not as a rule- 

 vanish. It will vanish either if the frequency be symmetrical about .<', 0, and 

 x a = 0, or if there be no correlation between the x. and organs, i.e., if / be of the 

 form /(a:;) Xf,(x;). 

 Fourthly, 



fl ,|, ,.(!' \0" / 7 7 / \ 



G,, = IJJ . / iLi .^ dx, dx, . . . (Ar,,, ...... (n.). 



This will not as a rule vanish. It will vanish, however, if the frequency of x. be 

 symmetrical about its mean. 

 Fifthly, 



(v.), 



all of which will generally be finite, but admit, like C ; ,.< and G, s , of calculation when 

 the form of the frequency / is given. Hence 



P A = Po expt. - 1 {B r (A/i,.)'- - 20,,, A/i, A/i,, - 2G,, Afc ( . Ac. 

 + E s (Ac s ) 2 - 2F M , Ac 3 Ac, + &c. . . .}, 



X expt. (terms in cubic and higher orders of the A's) . . (vi.). 



This represents the probability of the observed unit, i.e., the individuals 

 (Xi, x 2 , . . . x m , for all sets), occurring, on the assumption that errors A/i,, A/( 2 , . . . 

 VOL. cxci. A. 2 H 



