268 
Wf" 11 =@ (Ye $1, 2y2,t at a (1) 
which may also be thus written: 
d\? (aA4,)? , 
(s) YY. = rE RE Y 2° (1) 
The general integral required, may be thus written: 
2 t —l 
Iue= I= FEE GM YeottHn)i — @ 
an expression which may be developed into the sum of two 
series, as follows, 
at 
Yet =e. tos Fa Yeuiot aay Ae Yen yt eee 
, at? , ; ait? , 
So ro, at 1.2.3 ro Y x—1,0 + 1.2.3.455" Y 6-40 + &e.; 
(2) 
and may be put under this other form, 
Tv 
2 3 G 
Yx,t = — Boa Yet ah d8 cos (2/6) cos (2 at sin 8) 
Tv 
1 Sao 5 . ; : 
+ es Zn—w0 Yrtn0 \, d@ cos(2/@) cosec @ sin (2 at sin®); (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"y, , and 7/, .. 
Supposing now that these arbitrary forms or initial con- 
ditions are such, that 
T sins hie ie Tv 
Y~,o —nvers 2r -, andy’, , = —2ansin—sin2r-, (3 
Yx,0 n Y x0 n n 
for all values of the integer x between the limits 0 and —in, 
n and z being positive and large, but finite integer numbers, 
