CEUCES MATHKiMATlC.ïî. 17 



Equating coëlïicieuts of x". 



2 2 



^a„a,i n cvon. 



• . a„(2"-' — 1) = rt„_,a,-1-(T„.,<i.,+ .... »,+i«„_i, « odrl ; 



2 2 



^a„a„, w even. 



Now by equating coefficients of .r, of ./;', etc., we readily find that 



Assume that this hiw hokls up to the coefficient a„_i inclusive. Then, 

 "^ ^ n!l (n— 1)! l!^(«-2)!2! ^ 



—~ — n oven 



n ! , 



n od( 



n+1 , n — 1 



n even ; ""i 

 n odd. J 



_ 1 f C; + C; + . . . . JC;;, n even ; 



" 2 

 1 



= (2"--l),7, by (1) 



1 



• . ff„ — 



n ! 



and the law of the coefficients, holding good for a,„ is established. 



T X x^ t'^ 



'.a'^=l+:ji+2!+3-!+ ■•■■n\ + -- 



III. 



Expansion of the Sine and Cosine without the Use of Limits. 



Development of sin 6 in ascending powers of ^. 



It is well known that the only legitimate assumption for the expression of sin f^ in 



Sec. Ill, 1889. 3. 



