1886.] e x , log e {\±x), {l+x) m . 423 



Hence the equations 



1& + 2& + ... +r@ r = r-t 

 /3 t + & + ...+ &=r-s-l 

 have the same solutions as the equations 



la, + 2a 2 +...+(r-0«,-* = r-« 

 a i + a 2 + ■•■ +a r _ t —r — s—1, 



where 1 < t < s + 1. 



Hence coefficient of 7i r_s_1 in — is the same as in 



n 



c + c t + c 2 + ...+c r . 1 . 



Hence coefficients of all powers of n are the same in — - and 



n 



c + c l + c i +...+c r _ 1 ; 



rc r 

 .: — = c + c 1 + c 2 +... + c,.. 1 , 



whence - / —^ = c +c 1 + c 2 +... + c r _ 2 ; 



.'. rc r = (n + r—l)c r . 1 ; 



_n+r— 1 _n+r—l n+r—2 



• • C r — ■ C r _i — — . ~ z C„_o — • • • 



r ry r l m m 1 



_n + i — 1 n + r — 2 n 

 — r ' r -l '"1 C °- 

 But c = 1 ; 



_ n (n + 1) ... (n + 1 — 1) 

 c . 



T ! 



This demonstrates the identity of the two series in question. 

 Art. 3. The Logarithmic Series. 



X 3? X^ 



The series y + — + — + . . . is convergent if x < 1. 



Hence by the Exponential Theorem . 

 (?+* 2 + f+...) 



en 2 3 / 



- , (X x 2 X* \ 1 (x x' 2 x s V 



