302 Proceedings of the Royal Society of Edinburgh. [Sess. 
in which each component of E depends only on X, Y, and Z and is a solution 
of Laplace’s equation in these variables. 
To solve the equations (5) and (6), we notice that if e is a small quantity 
whose square may be neglected, equations (5) imply that the transformation 
x = x + eX, y' = y + eY, z = z + eZ 
is an infinitesimal conformal transformation of a space of three dimensions, 
for we have a relation of type 
dx 2 + dy' 2 + dz 2 = (1 + 2 e\)(dx 2 + dy 2 + dz 2 ), 
where s is kept constant. By a well-known result we may write * 
X = a(x 2 — y 2 — z 2 ) + 2f3xy + 2ya?z + px — ?z?/ + mz + u, 
Y = 2a xy + /3(?/ 2 — z 2 — ic 2 ) + 2yyz + nx +py — Iz + v, 
Z = 2a xz -f 2 j3yz + y (z 2 - x 2 - y 2 ) — mx + ly +pz + w , 
where a, /3, y, l, m, f ^ are functions of s. With these values of 
X, Y, and Z equations (6) are satisfied if a, /3, y, and p are constants, and if 
— =2 a, =20, 
ds ds 
These equations give 
l = Iq + 2aS, 
u = u 0 + l 0 8 + as 2 , 
dll _ 9 du _ 7 dv __ dw 
ds ^ ds ’ ds ds 
m = m 0 + 2/3s, n = n 0 + 2ys, 
v = v 0 + m Q s + /3s 2 , w = w 0 + n 0 s + ys 2 , 
where l 0 , m 0 , n 0 , u Q , v 0 , w 0 are arbitrary constants. The expressions for 
X, Y, and Z now take the form 
X = a(x 2 + s 2 -y 2 - z 2 ) + 2 f3(xy + zs) + 2y (xz - ys) +px — n 0 y + m 0 z + l 0 s + u 0 , 
Y = 2a (xy - zs) + f3(y 2 + s 2 - z 2 - x 2 ) + 2 y(yz + xs) + n 0 x + py - l 0 z + m {) s + v 0 , 
Z = 2a (xz + ys) + 2 f3(yz - xs) + y(z 2 + s 2 - x 2 - y 2 ) - m 0 x + l 0 y + pz + n 0 s + w 0 , 
and we have the result that a solution of Laplaces equation in the variables 
X, Y, Z is a solution of the wave-equation in the variables x, y, z, t. 
§ 2. A particular case of the last theorem is of special interest. If in the 
preceding equations we put l 0 = m 0 = n 0 = u 0 = v 0 = w 0 =p = 0, the equations 
resemble the formulae of Rodrigues*) - for a transformation of rectangular 
co-ordinates from X, Y, Z to a, /3, y, and it is evident that a solution of 
Laplace’s equation in the variables X, Y, Z is also a solution of Laplace’s 
equation in the variables a, /3, y. This result may be used to express a 
standard set of spherical harmonics for one system of polar co-ordinates 
(O', <fi') in terms of a standard set of spherical harmonics for another set of 
polar co-ordinates (0, <p). The formulae of transformation have already been 
* Sophus Lie, Theorie der Transformationgruppen, Bd. ii, p. 460. 
f See Whittaker’s Analytical Dynamics, p. 8. 
