96 
Proceedings of the Royal Society of Edinburgh. [Sess. 
These examples are sufficient to show the importance of the confluent 
hypergeometric functions of two variables in four-dimensional potential 
theory. We present them together in the following scheme, adding Appell’s 
previous results on Hermite’s two- variable polynomials, and the correspond- 
ing relations between one- variable functions and the three-dimensional 
potential theory : the scheme explains itself. 
Solutions of Laplace’s Equation by Hyperueometric Functions. 
(a) 3 Dimensions— One Variable. 
1. Sphere 
2. Parabolic cylinder 
3. Circular cylinder 
Legendre functions { P ^taVfa^on hyperge °' 
Weber functions { P^ticular case of the confluent 
f hypergeometnc function 4 s . 
Bessel functions 1 “"fluent hypergeometric function 
l -B. 
(6) 4 Dimensions — Two* Variables. 
1. Hypersphere 
2. Parabolic hypercylinder j 
3. Hyperparaboloi'd 
4. Spherical hypercylinder 
Hermite polynomials (Particular case of the hypergeo- 
r J ( metric function _r 2 . 
v f particular case of the confluent 
Function W k, n., v \ hypergeometric function ¥ 2 , de- 
t duced from F 2 . 
Function s, 1 confluent hypergeometric function, 
2 ! deduced from F 3 . 
{Issued separately May 9, 1921.) 
