Eoss — Verb-Fanctio)is. , 61 



Pirst, we must observe a point in the series for namely, that 



the coefficients of all the integral powers of (B, except namely, of 



n 2n 3» kn 



IS", ^-ir, . . . n , 



where k is a positive integer, all vanish. The general expression for 

 these coefficients is 



( n hi 



IS'ow, by the definition in § 16, (^)i„+i denotes the coefficient of the 

 (kn + 2y^ term of the expansion of 



But as h is a positive integer, and as has no term lower than^_„+ij8S 

 the expansion of can possess only hi - h + 1 terms. Hence the 

 {hi + 2y^ term must be zero ; and the coefficients of the integral powers 

 of except in the value of [t/'n]"^ must all vanish (but only when 

 the lowest power of /? in is not lower than p^). 

 Jfow, for brevity, write 



Then = t^a + + Lior^ .... 



Xr, = tih + toh^ + t_ib~^ .... 

 ^3 = tiC + toC^ + LiC~^ .... 



where ti is inserted for symmetry and = 1. Hence the sum of all 

 these series should give - p^i ; and the sum of their products taken 

 successively two, three, . . . ., at a time should give successively 



-i?_3, p-i, . . . .; 



and the product of all together should give + y. 



To study the question we can employ the notation used for 

 symmetrical functions of the roots of an equation, and write 



iSf„» = «'«+i- + c- ; 



and ^aH"" = aH' + a'^'c' -f h"'c' .... 



and so on. As 5, . . . . are the 7i^^ roots of a quantity 



I 



