381 
sc Tf, for simplicity, we represent the expression = 1 
by A, we find for « the values 
u = cos (wA?) f, (ys 2) + sin (vA?) A yen i(y 2) , Ad Mefsloi = 
+ cos (#A) f (ys z) — sin (7A?) ne pele 2) " =e z) 
and if we assume 
Ai Ys 2) +hY 2) =o 
Fi Ys 2) ~ fa (Ys 2) = de 
we find for the general value of w the expression 
od ala ts kp oon oie re : 
u = cos (#A®) ¢, + sin(wA®) A ey ae (III.) 
g: and ¢, being arbitrary functions of y and z. 
«« This solution agrees substantially with the one which you 
have obtained. If we develop the cosine and the sine, each 
of the operating functions will assume the form of a o. ex- 
2 d? 
dy ly® Ye? 
and the operation can then be performed when ¢, and ¢, are 
given. We shall, in fact, have 
u= F,+jF,+kF;, where 
1 do, apy as di, d‘d, d'g, 
shag lon + de :) * veri ih dx*dy? of 52) 
pressed in ascending integral powers of A, i. e., of — 
— &e. 
_,, Abs x (dp, 2 
Toeeigg © ine cal a + aa) * he 
_ _, Abs e (dd, dd, 
oo ae TS. ites W gmc | 
results agreeing with those which you have given in your 
paper of April 9th, 1855. 
‘‘ Instead of deducing these results directly from the sym- 
bolical forms to which the above analysis leads, your investi- 
gations conduct you to a process which consists in substituting 
