168 
and as these inverse operations are respectively, 
P P y 
EPID DS DT) 71 gXIDYkDy) oy 1D) a 
and 
erIDykDs) DP) -1 peo). or 7 Dy Tr, 
we shall have 
V= wD aw? Dy 2 U+ M, ] (y — jkr, a2 ke) + Mf, (y Tiley Zo+ kr), 
the two latter terms being the solution of (1). This com- 
plete solution, when developed, appears, in general, in the 
form, 
F,+jF,+kFr, 
F,, F,, and F;, being different functions of x, y, z, which 
singly satisfy the proposed equation. 
‘¢ For instance, we have seen above that 
EID HDS) y223 = yz — a (2° + 3y?z) + atz, 
+ 27 (ayz? + x°yz), 
+k (Bayz? — v2? — ay? + 145), 
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, 
CY Gv EV ave 
aot ape age 
in which U is a function of w, x, y, and z, may be reduced to 
the symbolic form, 
(D+iD, +jD,+hD,) (D -iD,-jD,-kD;) V=U, 
the solution of which depends on the inversion of the opera- 
tors, 
D+iD,+jD,+ kD;, and D-iD, -jD,-kD,. 
Putting 
iD, +jD; + kD; = Db, 
a notation employed by Sir William Hamilton, we shall have 
