MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 299 



, ., J.IW O V * L / > / A v 



'7a- ' x ' v- + (/ + 4) 



i/* + (> + 2)- 



_ a, 



n (r + 4) {v* + (r + 2)-} G (r + 2, v) 

 " 2 " + (r + 4) 3 G (r, i) 



or, 



i (r + 1) (r + 2) (r + 4) .. 



- " = ' -# ^ + <y + 4)2 



Precisely similar reductions lead us to 



f+ <P(logy) , nv (r + 1) (r + 2) ... . 



,.,= V 7 ~dx= rri r^ TST (cxxxvin.), 



J - X J dxda a- {v 1 + (/ + 4) 2 } 



- dx = r-; .... (cxxxix.), 



gy) 7 (/ + 1) (/ + 2) / i \ 



55-S-i ft T / rv I i 



rZv a j/ 2 + (r + 2)- ' 



\ *> i t"l / i^ j \ 



7? - - Mi.), 



(cxlii.), 



f '/) 1 ' I> (/' + 1) / ! \ 



; t/; ^ = ---~ 9 , (cxlni.), 



[ + * ''MlOg'/) , ' M f~i , M / V \ 



M s= M- ^dx= n {logG(r,v)} . . . (cxliv.), 

 ]__ (/?- "'" 



- rfe = n {log G (r, v)} . . . (cxlv.), 



= n y- s (logG(/', i')} . . . (cxlvi.). 



It will now be needful to find easily calculable series for the second differentials of 

 log G (r, v). These can be obtained from (cxxxiv.). We find 



^ (log G (r, v)} = - + log cos e + '^ r - 



_ o B a.+i(~ !)' < l _ 2 2"+ 2 cos 2s+2 <f> cos ("2s + 2) <j>] . . (cxlvii.), 

 (2s+2)^- { 



d ., ~ , ,,, , sin <f> cos <f> , 



-r {logG(r 5 ^)}=A7r- 



r 



2 2s+2 cos"*^ sin (2s + 2) ^ . ' . . (cxlviii.). 

 2 Q 2 



