MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 285 



It is clear that if w, and m., be at all large, which they frequently will be, we may 

 omit the series S, or even reduce e,, e,, e 3 to l/m^ 1/m.f, and l/(m, + m. t + 1)> 

 respectively. 



Making use of (Ixxxvii.) and (Ixxxviii.) we easily find 



... = n6,., = - 



a, 4 = H. U 



a,, = w,, = - 



The next stage is to calculate the determinant 



6,,, 6,,,, 6, 3 , 6,4 



6,3, 6,3, 633, 6 34 



6,,, 6,4, 6 31 , 644 

 and the minors B,,,, &c., of b a -, &c. We shall then determine 



S * = V A 7 ' S ' : = T A 7 ' 2 '"' = T A' ' "" 



fy g^cfa= -f (M ' + ""X"" t "^ 1> ...... (xcvl). 



Jo" 7 tfoao &- ?/;., 1 



f < P (log ;/) 7 w )M, 4- m, + 1 / \ 



?/ ,-f ax = ----- - .......... (xcvn.). 



Jo dxdm l I) tiii 



6 if (logy) . n (w, + at., + 1) 

 / -- - v --- cto =-- - ......... (xcvni.). 



o dxdm., I M,, 



rf 2 (logw) , ra / ,- /", + 1 / \ 



( ^ ^ ='j s ("i + '"^ + 1) -; ! ( x " x -)- 



(^(log.//) , // , 



^bd^ dx '- 



d- (log v) , n MI + 1 , . . 



^^! dx = -T~^~ ............ ^- 



d*-(logy) , x y .. K 



' fa J> dx = n (e. - e,) ............. (cii.). 



