345 
4. aaa , 
therefore we suppose that the initial data c.., 1,0 and O. 1,0 
are all such as to agree with this particular solution, that is, 
if we have, for all values of g and h, 
eee ‘ Reoe ar . 
oe hy 0. > 1 An cos(e 1 Ziyi % i)? (14) 
4 pe Ee Se tear, : ee ny . 
Fh 0= 5X ,4,,8In(€, Ziyi % i a (14) 
we see, @ priori, that the multiple integrations ought to 
admit of being all effected in finite terms, so as to reduce 
the general expression (9) to the particular form (4); an 
expectation which the calculation, accordingly, @ posteriori, 
proves to be correct. An analogous but less simple re- 
duction takes place, when we suppose that the initial 
equations (14) and (14) hold good, after their second mem- 
bers have been multiplied by a discontinuous factor such as 
(1 fa 2( sin(k3,,i0 # ) a), (15) 
To i 
which is = 1, or = 1, or = 0, according as the sum 
Lo”: x, ; is < 0, or=0, or>0. It is found that, in this 
case, the 2n successive integrations (required for the general 
solution) can in part be completely effected, and in the 
remaining part be reduced to the calculation of a simple 
definite integral ; in such a manner that the expression (9) 
_ now reduces itself rigorously to the following : 
, \ ‘ ‘ ‘ 725% | 
Be ht = BX 14g CO8(8y + t8y — B45, 5) | 
a (16) 
12g dk : yeahs 
+55) pop (coe, +m, sin €,); 
in which 
L, =P kcos kx —@ ksinka, 
(17) 
M, = Pk sinke + Qk cos ke, 
