96 Mr. J. O'Kinealy on Fourier's Theorem. 



(where i is cipher or a positive integer), or 



The original differential equation becomes thus 



M D * + ^X D *- 2 -^) sin/w=0 ' 



or 



*t \ A , A ^ X , A ^ lra; , 



/(#)=A-j-A,cos— — hA 2 cos — — h. . . . 



A» A/ 



-f B, sin -^ — \- i3 2 sin - T - .... 



A A< 



rrA + SX-COS? 7 ^ 



2=1 X 



*"=" . 2irix 

 + zB { sin » 



2 = 1 A. 



This is Fourier's theorem ; and, determining the constants in 



the usual way by integrating between and \, and by multi- 



2nrix 2tt'ix 



plying by cos — — . sin — — and then integrating, we get the 



usual form, 



jfi \ 1 C\ f a 7 2%* 27rix C K r> s 2irix 1 

 /W = > \ fi x ) • ^ + =- S cos — — - I /(a?) cos-—- ffo 

 V fl A ^=i A Jo x 



2%« . 2irix C x „ n • 2irix , 

 4- - 2*sm — — i /(a?) sm — — a». 

 A t - = , A J A. 



In the same way we can obtain other forms of Fourier's theo- 

 rem. If f(x) =f(% + h)j we have generally f{x)=f{ps-{-nh) f 

 where n is an integer; or (e nhDn —\)f(x) = 0, which gives the 

 same solution as above if we put n\ in place of A. 



Hence we find 



m = ^F /w dn + ^r mdn • c ° s s • x c ° s ^ &c - 



=.t j(x)dn + — 2, cos — — - • 1 /(a?) . cos — ^— dn 

 n\J Q J > n\ i=1 riX J o yw n\ 



2 % x . 2*iriac f »* x • 2wwr . 

 ■f — - . z, sm — — - . | fix) . sin — — dn, 

 rik i:=1 nX J n\ 



The above method of solution may be applied to other some- 



