MR. HEARN ON A CLASS OF PARTIAL DIFFERENTIAL EQUATIONS. 
135 
=D^r, and is therefore true for u— 0. Under these restrictions it will therefore be 
d 
true for any positive integral value of n. Now the symbol D represents e~* and 
therefore if U=0, D”U=0, so that we have 
+p,zDv,=0, 
or 
or 
d^v n . fi n ' dp ' — q dv n — 
dxdy Afi n dx e dy ’ 
d dv n 
dx dy fi n dy ' 
dv n Afin dx ’ 
dy 
integrating with respect to x, 
dv n -- 
— =e Afin X y 
fin 
and 
e Afin xydy+^x, 
u= J)~ n v n — e * J' e* v n dydy .... 
the integral sign repeated n times. The theorem is therefore demonstrated. 
It may be easily shown that the equation of Laplace’s coefficients is included in 
the class here considered. 
The equation of Laplace by a proper choice of independent variables assumes the 
form 
d% n .n + 1 
dxdy' a? U 
* 4 cos' 2 
Hence with reference to the preceding investigation, 
D=cos 2 
Hence 
Hence 
MDCCCXLVI. 
9 y—x d 
\ — - • — 
and a„ = 
n.n+ l 
dy 
, oV~ x 
e _0__;Cos2£-__ . 
dp y—x 
s = — tan — 
d q <p , 1 
d^y+2 e, = 0 - 
c= i- Also A«*„= ; 
(3 n —n and Aj3„=l. 
A rv 
Acc n —C= -X ; 
