88 Proceedings of the Royal Society of Edinburgh. [Sess. 
Introducing again the W k,^, v function, we shall form the system 
satisfied by 
There is no need to give the complete calculation : let us say only that, 
adding together the equations thus obtained, we find that 0 is a solution of 
r + t + -p + ‘ 1 q + z(-+ t)(i- V)|o, 
x y \x 2 y V 
y \x* r 
the parameter k having disappeared. 
/ Yb 
If we take therefore 4/x 2 — \ = n(n-\-l), or = ^ + we obtain complete 
identity with equation (C), so that 
U Ju, = W, « +1 n - . ^ 
2 +?> 2it ? V2 
is a function of the hyperparaboloid of revolution. 
We shall now make several remarks touching these functions. 
1. The confluent function which appears in the question is of a type 
we studied in Part I, Chapter Y, where the two last parameters jx and v 
are of the form - + J, a being an integer. Referring to a property estab- 
lished there, we can say that the function of the hyperparaboloid of 
revolution is expressible in terms of the parabolic-cylinder functions. 
2. Let us consider the change of variable 
O 9 
x = — - — sin (p cos 0 
2 
oi2 
y = - — - — sin <£ sin 0 
u 2 — v 2 , 
Z = COS (f) 
t = uv. 
The hypersurface u = const., 
i(x 2 + y 2 + z 2 ) = (u 2 
is obtained through rotation about the £-axis of the surface 
4(« 2 + y 2 ) = (« 2 — ^) 2 , 
which is of the fourth order in the xyt space, and itself the result of the 
rotation about the £-axis of the parabola 
