MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 247 

 d(logz) _ (logR) d /E^\ d 



~ 



Differentiating the first of these again with regard to o-,, and summing for all 

 possible values of x's, we find 



ll = 



O", \ It ]' 



But 

 Hence 



:1 /I! -L I> V 



(xxiv.). 



Differentiating (xxii.) with regard to cr_, and summing, we have at once 



log 2) , /' r,.,!!,., 



< ix, doc., . . . dx = ,, = - . . . (xxv.). 



r, ((&., <7,o-., 1 1 



Differentiating (xxii.) with regard to i\., and summing, we have 



rrr rf*(logs) , 7 , // [ // / R \ '/ /i{,/\ 



z T^fc" (/Xl f/; ''- ' ' t/a! ' = ]Cl - = 7" P + k"'' 1 - JT PF / 



JJJ na' l nr t ., & l ["/V> \, K / "* ii \ " ' 



A 



o-, l^A K 



or 



,c r _, = nl^o/o-JI (xxvi.). 



Differentiating (xxii.) with regard to r. fi and summing, we find 



n d E,, n 



__ 

 ^ dr a H ' <r, rfr. i \ E 



i rf /E.. + S. (,.,.)' 



... \ E 



or 



,c a = (xxvii.). 



Our next step is to differentiate (xxiii.) with regard to r, 2 and sum. We have 



