Schubert' s investigation of Eejiler^ 8 Problem, 



17 



dependent of 



The general term of sin ?e (E) (§ 



neglectin- the coefficients by which they arc multiplied, has 



llowin^ form, e"sia {n±i 

 (§. 13), it assumes the form 



d 



d by 



» 



fi + sin {n±i—2r) ^ + etc.), or e" (l+eHeV.-) (sin a/* + sin (a— S).« 



+...)? which is al 



form ofe(D)(§ 



The 



dent, that in every term e" sin m^ of the series (A) 



a 



m, or a=m-h2, or in general 



m-fS«, in otlier words, a 



d 



m mnst be either both even or both odd, and m can never be great- 



er than a : 



I shall hereafter suppose 



a 



S s, or 



a — m 



2 



the letters denoting any affirmative number, not exclud 



ing 0. If we were to give to w a value not falling under the form 

 a— 2s. the term Ae" sin m^u would be impossible, or A=0. 



2 



e + j^' sin it (A) (^ 3), the quan- 



§ 19. Since v is in general 

 A will depend in part upon e, and in part upon 



The 



former part, which we shall call A 



must be deduced from 



the general term of e (D) (§ 9)= ±^. ~ (w-?r)-sin («-gi>, 



On^V I 



which will agree with the proposed term Ae" sin ma, if we make 



*ii 



n 



a, and n — zr—m, or r 



«> 



a — m 



2 



s (§ 18) : whence we set 



© 



AW 



± 



1 



■j;;^\-2) J N denoting the general coefficient of a bino- 



mial raised to the power n (§ 



I 



rmal 



A for the general coefficient of .a binomial 



raised to the power a, so that Ao 



1,A, 



fl, A3 



a{a — 1) 

 1. 2 



and 



A, 



<g— 1) (g—s+l) 



1. 2 



s 



we shall get (&) A<"^ 



± 



A, 



m 



1.2.,.^\2 



