1886.] e x , log, (1 ± x), (1 + x) m . 417 



Art. 1. The Exponential Theorem. 



(a) Let f(m) denote the series 1 + m + ^- + . . . + — + . . . 



It is required to show that whatever m and n may be 



/Hx/(n)==/(m + »).. (A). 



Let the coefficient of m r n' be calculated on both sides. 



..11 



On the left hand it is — : . — , . 

 r ! s! 



On the right hand the term mV can only occur in the term 



— . Its coefficient is therefore . — , — ,-' . 



r + s ! r + s\ rl si 



The coefficient of any term mV being the same on both sides, 

 the equation is demonstrated. 



(/3) Now suppose x a positive integer, then 



[f{l)Y =/(l) x/(l) x/(l) x ... until there are x factors 



=f(x), by repeated use of the equation (A). 



V) 



(y) Next let a; be a positive fraction - , where p and q are 

 positive integers ; 



But/f "j x/(-j x/( -) x ... until there are j factors, 



—f(p)> ^ repeated use of the equation (A), 



= [/(l)] p , by the previous case since p is a positive integer; 



fp y 



/(|)J =[/(!)?, 

 /(f) =[/(!)] I 



<qj 



••• /W =[/(!)?• 

 (8) Lastly, let x be negative and = — n, so that n is positive. 



Then /(— n) x/(n) =/(0), by equation (A), 



= 1, 



•■•/(-») =7^; 



VOL. V. PT. VI. 29 



