222 
and in F, the coefficient of x?" is 
(1 dfa@ dyn 
(2n +1)! dz Lay” dz| f 
‘“* With respect to these expressions F,, 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, dF, dF, dF, dF, dF, 
dz dy’ dz dz’? dy dx’ 
in virtue of which, the expression 
Fidz + 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 
ae | D, F,. 
(D3 + D3 + D3) F-[ D, (D3 + D3 + D}) F,=0, 
and a similar proof applies in the case of £;. 
“5, Writing f£ in place of “ in F,, or a in FP, we see 
that 
F,=xfp- 
2h), 2 (16, , Of wh) a 
n(Gee ee aa Fad age 
will be a solution of Laplace’s Equation, whatever function 
of y and z is denoted by /:. 
“¢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- 
