Conwav—The Partial Differential Equations of Mathematical Physves. Wit 
the same meaning for the symbols as before, we can verify the following 
solution :— 
S22) (AS an Au?) )(n Ss 3). 
The third and general type of solution is as follows :— 
1 
Suppose the variables to consist of m groups 21, Uo, ... 2'n'3 U1, Uo, --- Un"; 
- oi”, a”... ¢™,~™, and that the equation is so written that the products of 
operators out of different groups do not occur. If @’,, be the coefficient of 
ad 
amour) 
the following is a solution :— 
f L(y)! 1 on (iS Q” £3n’—m—2 
I11.—Equations or Crass I. 
The singular solution of 
De en nee 
CBr GBs OG 
is, by what has been shown above, 
1 2 72 t ~ 2\-3(72-2) . 
(air te BF eo of o tt Gy SP ; 
this is infinite only at the one point—the origin, and is there infinite to degree 
n — 2. 
IV.—Eeuarions or Crass II. 
The solution in this case will be of a different type, according as u is odd or 
even. As the first is far simpler, we shall take it first. We shall also defer till 
afterwards the general solution, and shall begin by finding a solution which will 
be infinite for the value zero of the variables 7, %, ... 2%, and for all values of 7. 
A solution of this equation is 
f(u)[ae + a+... 4%, —(t-uy rr, 
where w is any constant. The sum of any number of similar terms is also a 
solution, provided that the path of summation is independent of m, %, ... %, or @. 
