198 Conway—The Partial Differential Lquations of Mathematical Physics. 
If a vanishes, the type of the equation is 
ep 2 Dic: onenre @ 
EPL LON SG a oe 
Pg 
Oa? Oa, " Ga? Gy Gore 
OY n> 
In a similar manner, we get 
dy 
(7.52) | Pca 
¢=p le = vy) ‘ 
2 
If n is even, we take the integral surrounding o and 7 and if 2 be odd, we 
Vi 
take a contour about an Hence, 
2 
4 
p = const. pun) ye) PF 
From this we can get different solutions, as follows :— 
ra 
d = pa”) en |(y AS ye) BID tp(y-) dv. 
The integral is, of course, a Bessel function. 
VII.—Non-SyMMETRICAL SoLurtons, 
All the above solutions belong to the equations 
op Dal Oy _ @ 
or? - @p  ° 
op m-1op _ oY 
or? yr Or ot?’ 
> n—-1p 8h m—1 2% 
ALD + | B —— © 
or T Or Op° p °p 
We can get non-symmetrical solutions as follows:—For example, in the last 
equation: let S,, =, be homogeneous functions of integral degrees p and g, respec- 
tively, In 2%, %, ++. %j Yr) Yo) ++ + Yn, and let them be solutions respectively of 
ad ad 
ee ay a 
and of 
ad ad 
OY,” OY 
