268 



which may also be thus written : 



/ay _ (+*.y (iy 



The general integral required, may be thus written : 



* * - fi ■•- #^ (s « ^ 2 i ~^ • + - & •> ; (2) 



an expression which may be developed into the sum of two 

 series, as follows, 



aH 2 a 4 1* 



and may be put under this other form, 



IT 



2 °° f 2 



y», * = - s w- oo & -i- * o X de cos ( 2 ^) cos ( 2 «* sin ) 



+ — S W-I 2/' M .,,ol 2 ^cos(2^)cosec0sin(2^sin0); (2)" 



the first line of (2)' or (2)" expressing the effect of the initial 

 displacements, and the second line expressing the effect of 

 the initial velocities, for all possible suppositions respecting 

 these initial data, or for all possible forms of the two arbi- 

 trary functions"^ and y x> . 



Supposing now that these arbitrary forms or initial cou- 

 ditions are such, that 



Vx, o = n v ? rs 2x ~> and V'x, o = ~ 2 a ? sin ~ sin 2x -> ( 3 ) 



for all values of the integer x between the limits and —in, 

 n and i being positive and large, but finite integer numbers, 



