MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 267 



(16.) The first stage in the investigation is to apply the general proposition of our 

 Art. 2, to 



logy = log n+ logy- (p + l) + plog(p + 1) logr(p -f 1) 



We find : 



if- (log//) i> 1 



-- ''- 



... , 



^ <? + 



1 



(1 + '''A') 2 /' 



civ dp a \1 + ;>/" 2' + 1 



c]-(lo S >,) __ P _L . t 

 da; (fry p + 1 (I 4- </")- 



^ y) =-^ lo r ^+ ') + ,,+ 11^ 



f r-(io g //)_ 

 " 





P. -\ 



i vi -- / ^ __ -r_ ' ~ .__; i 



Let I, = Vi ( 1 4- --Ve'^daj, then we easily find I , I , and 



J J J-\ / i" 



(P + 



^ 



By aid of these we can at once write down the integrals of the above expressions 

 multiplied by y, since n = I,,. We find with the notation of p. 243, 



r , , .. r ^ ( lo s y) ^ _ "T' J 



'ti 



f 

 _, 



tfxdp PCP-I)' 



C (P (log y) j 2n 



y ~-~ dx= : , 



J _ * rfa; f?7 p 1 



(V (log y) . Iff-, . . -2 



- 



a 33 = 



- r_' 



2 M 2 



