306 Proceedings of the Koyal Society of Edinburgh. [Sess. 
Now it follows at once from (8) that 
. _ i y+p r(s + i)r (« + s +p')Y(s + a — i)r(c) 
p K ' r(i? + 1 )r(8 -p + i)r(a + *)r(p + a - c + i)T(c + sj 
An interesting expansion is obtained by writing rj = 1 — £ 
It should be noticed that the above theorem enables us to transform 
a series of type 
oo 
2 C S F( - s, a + c, £)F( - s, a + s, c, rj) 
into a series of type 
2D^l-f T «/)'P(-P.« + Re.Zz$ri} 
p = 0 ' ^ ' 
The problem of expanding an analytic function / (f ) in a series of type 
M) = 2 MX _ p> a +p> '■'< 0 
P — 0 
has been discussed by H. A. Webb.* He finds that if the real parts of c 
and a — c + 1 are positive, the series may represent the function within a 
region in the complex plane bounded by an ellipse whose foci are at the 
points 0, 1. 
§ 4. The result obtained at the end of § 1 may be generalised as 
follows : — Let us consider the transformation 
X = {x Y x 2 + -- y $ 2 - z x z 2 )x + (x Y y 2 + x 2 y x + 4% + z 2 sf)y + {z x x 2 + z 2 x x - y x $ 2 - y 2 sj)z 
~ (y i z 2 - V2 z i + ^i 5 2 - ^2 s i) s > 
Y = ( X lV2 + X 2V\ ~ Z 1 S 2 ~ Z 2 S l) X + (V$2 + S 1 S 2 - X 1 X 2 ~ Z \ Z 2)V + (Vl Z 2 + V2 Z l + X l 8 2 + X 2 S l) Z 
~ (Zi x 2~ Z 2 X I + Vl S 2~ V2 S l) S > 
Z = {zjX 2 + z 2 x x + y Y s 2 + y 2 s x )x + (y x z 2 + y 2 z Y - x^ 2 - x 2 sf)y + (z ± z 2 + - x Y x 2 - Vl y 2 )z 
~ (^ 1^2 ~ X 2V\ Z L S 2 ~ Z 2 S l)‘ 
Then, if s 1 = — ict v s 2 = —ict 2 , it can be proved that if V = F(X, Y, Z) is a 
solution of Laplace’s equation in the variables X, Y, Z, it is a solution of the 
wave-equation in each of the three sets of variables x, y,z,t ; x v y v z v t x , 
^2 > 2/2’ ^2’ 4* 
Adopting Whittaker’s expression t for a solution of Laplace’s equation, 
V = f /(X cos e + Y sin 6 + iZ, $)d$, 
* Phil. Trans ., A, vol. cciv (1904), p. 481. 
f Monthly Notices of the Royal Astron. Soc., vol. lxii (1902), p. 617 ; Math. Ann., Bd. lvii 
(1903), p. 333. 
