218 
‘«¢ Now the symbolic form of Laplace’s equation which was 
integrated was 
(D, -tD, -jD,) (D:+tDz+jD,) .V=9, 
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, 
(t(D, +jD,+ kD.) .V=0. 
Now, if we confine our regard to this latter form, and substi- 
tute for the imaginary symbols real quantities a, 6, ¢, it can 
readily be shown that the solution of the equation, 
(aD, + bD,+cD.).V=0 
is 
b 
where % and v, are arbitrary homogenous functions, of the 
order zero, of the quantities respectively under them, or 
2 peared AE: Se halted : Ath ec 
v-(F+$+2\o(§ e i¢éia a ae ae ce aa i). 
Ifnow, in the right-hand member of these equations, we replace 
the real quantities a, 4, c, by the imaginary symbols i, j, 4, 
respectively, we get 
x ed aa ¥ z= 
ee (erat) » Uy (€2, @, €°) +, (67, 6%, e ), 
ee ee 2 A ee 3 
(F+4+Z)-m (6,2, el) (6,6, 04), 
and 
Micah Py gt AOA 2) eee 
G47) 9G Bhovi eG kin elt i} 
and modifications of these analogous to that éstablished 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. 
