86 
Proceedings of the Royal Society of Edinburgh. [Sess. 
A solution may be obtained by taking 
U = U 1 («a, v) cos m<f) cos nt ; 
the function \J 1 verifies then the equation 
0 2 lL 0 2 U, 1 0IL 1 0IL 2 ( 1 1 \ TT 2/ 9 2 x TT A /m 
which we shall name the parabolic hypercylinder equation, the two-variable 
function U fu, v) being a function of the parabolic hypercylinder. 
It is obvious that equation (B) may be solved by the product of a 
function of u alone by a function of v alone : but solutions of that kind we 
shall consider as degenerate, and therefore reject them. 
A real two-variable solution, however, is easily found by considering 
the W fc ^ v system. If in this general system we take 
Jc = 0 
and replace x and y by we shall see that this function verifies 
p ( n 2 x 2 n 2 y 2 , 1 — m 2 \ ~ 
r-t-ny q + ^- r -^ + _j = 0 
if)-*- 
and that, if z x is a solution of this system, the function 
verifies 
1 --(*2+2/2) 
= — e 4 Zj 
xy 
r + p( nx + -) - nyq + z( - n 2 y 2 - 
s 
r + t + 
XJ \ XT 
+ q(ny + - nxp + z(^ - n 2 x 2 -~^j = 0, 
and therefore the single equation obtained by addition of these, i.e. 
- + ?- m 2 (~ + ^)z - n \ x2 + y 2 )* = 
x y \x z y z J 
which is precisely equation (B). We have then this remarkable result: 
the function 
1 — ~(w 2 +i? 2 ) / nu 2 nv 2 \ 
Vi(u, v ) = ~ c e VV o, ™ ~2 ) 
is a function of the parabolic hypercylinder, and Laplace’s product in 
parabolic hypercylindrical co-ordinates may be taken equal to 
7 T cos md> cos nt -% 2 +« 2 ) w ( nu 2 nv 2 \ 
U= * e 4 W 0i « 4— , — J. 
uv 2 2 \ l A / 
