168 
and as these inverse operations are respectively, 
Z gts DerkD3) J) e™ jDytkD3) or 72D: Tr 
and 
e jDkD3) D3 eu SEED): or 7 Dy Ww, 
we shall have 
Ve cD? Dy a U+ Mf, (y —jar, z-he) + Mf. (y+ jx, +h), 
the two latter terms being the solution of (1). This com- 
plete solution, when developed, appears, in general, in the 
form, 
Fi, + jP,+kF, 
F,, F., and F;, being different functions of x, y, z, which 
singly satisfy the proposed equation. 
‘«¢ For instance, we have seen above that 
GIP Ds) y2z28 = ye — 2 (2 ES 3y°2) + xz, 
+ 27 (ayz® + xyz), 
+k (3ay?z? — vz? — ay? + t2°), 
It will be found on trial, that each line in the right-hand mem- 
ber will by itself satisfy the equation of Laplace’s functions. 
‘«¢ The conclusions already obtained may be further genera- 
lized. For the equation, 
GV av gl ee sh Aa 
dw da dy dz’ 
in which U is a function of w, «, y, and z, may be reduced to 
the symbolic form, 
(D + iD, +jD, + kDs) (D -~ iD, -jD, = kDs) V= U, 
the solution of which depends on the inversion of the opera- 
tors, 
D+tD,+jD,+ kD;, and D-iD,-jD,-kD,. 
Putting 
iD, +jD,+kD;=P, 
a notation employed by Sir William Hamilton, we shall have 
