196 Conway—TZhe Partial Differential Equations of Mathematical Physics. 
V.—Eaquations or Crass III. 
We have now to consider the equation 
0? C2 a e? 0” C2 
aie PORN) peg a ary a SG res 
Oa? 049" OLp Oh OY OYm 
Let a denote any solution of 
CO) ob Ob 
ae oe Mane = Bype 
and 8 denote any solution of 
ob or op 
‘ Oy” We BY mn win ‘ 
and f a function of uw and v; then the result of substituting a8f for ¢ in the 
equation will be 
a 2B 
JB ap ou 7 ape? 
si 18 5a 88 be) ~ aa te ~ Ban) + 28 (Ge ~ ae) 
Hence, if we take f to be a function of w + 2, and a and B to be singular 
solutions of the respective equations, then the solution will be 
[|dudvaBf(u = v). 
If mand are odd, and if we put 
I? = gt te. En) Pp? = Ye Yor ee oe te Ine 
This solution will consist of a finite series of terms of the form 
i PASC = p)p 
