222 
and in F; the coefficient of 2?" is 
WR hice ibaa a 
(2n +1)! dz | dy? + ef ie 
“« With respect to these expressions Fj, F,, Fs, the follow- 
ing circumstances deserve notice :— 
‘1. They are entirely freed from imaginaries. 
*©2. Any one of them is a solution of Laplace’s equation. 
««3. They are connected together by the relations 
dF, dh ay dB, dhy_ dF. 
dz dy’ dx dz’ dy dx’ 
in virtue of which, the expression 
Fidzx+ F,dy + Fydz 
is an exact differential. 
“¢4, From the fact that F, is a solution of Laplace’s equa- 
tion, it follows, that F, and F, are likewise solutions. For as 
F-| DF. 
(Di+ Dis Di F=[' D, (Dis Di+ Dd Fix 0, 
and a similar proof applies in the case of F;. 
“5, Writing jf in place of ie in F,, or # in FP, we see 
that 
F,=axf,- 
Pf, Pf # (df. af df 
nies y = aM © CD ia) ae 
will be a solution of Laplace’s Equation, whatever function 
of y and z is denoted by /f.. 
“°6, If we add this value of F, to F,, we obtain a solution 
involving two arbitrary functions. It is exactly in this form 
that Lagrange has presented the solution of Laplace’s Equa- 
tion in his ‘‘ Mecanique Analytique,” p. 520. 
«7, It appears, then, that we are able to deduce a complete 
solution of Laplace’s Equation from one of the arbitrary func- 
