1920-21.] Hypergeometric Functions of Two Variables. 
87 
Chapter II. 
THE FUNCTIONS OF THE HYPERPARABOLOID OF REVOLUTION. 
Let us now, in like manner, consider the change of variables 
x = uv sin 0 cos 0 
y = uv sin 0 sin 0 
Z = UV COS 0 
u 2 — v 2 
2 
The hypersurfaces 0 = const, and 6 = const, are hyperplanes; the equation 
of the hypersurface u — const., obtained by elimination of v, <p , and 0, is 
x 2 + y 2 + z 2 = u 2 (u 2 — 2 1). 
It represents therefore an hypersurface of the second order, obtained by 
rotation about the £-axis of the quadric 
x 2 + y 2 = u 2 (u 2 — 2 t) y 
which is, in the xyt space, a paraboloid of revolution about the Laxis. We 
shall term this hypersurface an hyper paraboloid of revolution. An hyper- 
surface of the same type corresponds to v = const. 
Laplace’s equation is now 
, „ • , 0U1 . 
u 2 v z sm 9 — + 
CU J 
dv 
' o 2 • , 0U1 0 
U l V 2 Sill 0 — — I -f- 
^ dv J 00 
( u 2 + v 2 ) sin 0 
0U 
00 . 
+ 
00 
U 2 + V 2 0U 
_ sin 0 00 
p 
We easily get rid of the 6 variable by taking 
U = U-^w, v, 0) cos mO, 
wLich gives 
0 / o 2^l\ . 9 \ . / 2 . 2 \ 
— U 2 V 2 — -I + — U 2 V 2 —± ) + (u 2 + V 2 ) 
ou\ ou J ov\ dv 
1 0/i ,0U T \ m 2 IL 
_sin 0 00 \ ^ 00 / sin 2 0_ 
= 0. 
But it is also obvious to take 
p = P(“> -)P:(cos <£), 
where P™ is the associated Legendre function, which satisfies 
P»+ 1 )-PL1 P = 0 > 
sin 2 0 J 
and we obtain an equation in u and v only, 
1 d l . ,dP\ , 
— ( sm 0 — + 
sm 0 dcf)\ a0/ 
0 2 tL . 0 2 U O . 2 0U O . 2 0U 
+ 
+ - 
(C) 
du 2 dv 2 u du v dv 
a solution of which, really depending on two variables, will be a function 
of the hyperparaboloid of revolution. 
