381 
a 
by A, we find for w the values 
u = cos (vA) f, (y, 2) + sin (wA}) A hes) + ene) 
2 = 
‘Tf, for simplicity, we represent the expression 7 
+ cos (wA4) f; (y, z) — sin (vA) At a (ys 2) oes a 
and if we assume 
Si (Ys 2) + Fe 2) = hr 
fi(ys 2) — fa (Y> 2) = 9 
we find for the general value of u the expression 
= cos (vA) gi + sin (vA?) A% ve a a (III.) 
@: 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 ofa series eX- 
2 2 
pressed in ascending integral powers of A, i. e., of ie 4 > 
and the operation can then be performed when ¢, and ¢; are 
given. We shall, in fact, have 
u= F, + jF, + kF;, where 
2 2, 2 4 4 4 
Fi o.-79( Ge Nera : resale 1p da 2 + oe) 
dyy dz 1.2.3.4\ dat du*dy? dy* 
i — &e. 
- dp. _ = pe d*o, 
hbk iy alae dydz + &. 
_ , Ube a (do. Bos 
ae 1.3. “s(t i ae) 
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 toa process which consists in substituting 
