218 
** Now the symbolic form of Laplace’s equation which was 
integrated was 
(D,-iD, -jD,) (D,+iD,+jD,) .V=0, 
which is obviously unsymmetrical. It appears then possible, 
that in order to have arrived at a solution susceptible of useful 
application, we should not have taken this form, but one 
purely symmetrical, and such was pointed out nearly nine years 
since by Sir William Hamilton, namely, 
(iD, + jDy + kD;)*.V = 0. 
Now, if we confine our regard to this latter form, and substi- 
tute for the imaginary symbols real quantities a, b, c, it can 
readily be shown that the solution of the equation, 
(aD, + bD,+cD,)?.V=0 
is 
Lye EON ys EE lw Le Sees iE tied: 
Ke cenite 1 Up (6%, &, 6°) + (67, 65, e), 
where # and are arbitrary homogenous functions, of the 
order zero, of the quantities respectively under them, or 
DEY WON. sl Yah eae etter csi Pe 
a= =- = — er at) — = Shs mee Pe abst ee oS 
Gees 6 O28 a sae ce) ad bf 
Ifnow, in the right-hand member of these equations, we replace 
the real quantities a, 6, c, by the imaginary symbols 3, j, &, 
respectively, we get 
i 1D EN Seo eg 
( +o4 i) Uy (67, @, e*) + vy (e?, 6, e*), 
and 
zy ,?\—(¥_22 22 9) gfy 22 2a y 
G4+z)o(§ PROT a: ae Bk? i 7 
and modifications of these analogous to that established so con- 
clusively by Professor Graves, for the previous form, will 
give, I think, solutions of Laplace’s equation, which will sa- 
tisfy the conditions required. 
