288 

 where the second member may be reduced by means of" the relations 



X — p 



Therefore 



dx' ^^ ^ 2 x—^3 ^ 2 x—i3 



and 



dx* 2 J X — ,i 2 J X — ^ 







or, according to (3) 

 dhi 



d. 



-— + ?< c=r (n + /^) . 



This differential equation holds also if' p =:. 0. 

 Now the general integral of' this equation 



[n^p) \sin{x- 



T I ■i\ 



u =r A dn X -\- B cos x -\- (n -\-p) I sin {x — ^i) d^i 







gives the required value of u, when the constants A and B are 

 determined by the conditions 



Ju 

 x=zO H = — z= 



dx 



du 

 x = u — — = 1 {n ^ p = 0). 



dx 



Thus we obtain generalijn 



) I sin 



Iu+, (^) 



u = ('* + p) I dn [x — ,i) d^ 







and when n^= p ^0 



XL ^=z sin X. 



Introducing now the known expansions 



sin {x-i) := 2 [I, {x -,?) - /, {x-3) f I, (x-^) 

 and 



sinx = 2[I,ix) — I,{x) + I,{x) ..] 



we have according to (3j 



