LIQUID SPHEROID OP FINITE ELLIPTICITY. 
193 
The angular coordinate (f) is the same in both systems; but, except over the surfaces, 
p, will not be equal to jx, nor will any other two of the surfaces v = constant and 
^ = constant coincide. 
Put ju,' = cos 0. On transforming to {6, v, </>), equation (9) becomes 
3 [/ 3 1 L ^ 3 / • I — cos^ $ d'^Jr 
dvj'^ sin e dd ^ 30/ (v-- 1) sin3 6 dcf>^ 
0 
( 20 ), 
of which a solution, finite and continuous at all points within the spheroid (14), is 
xfj = (/) T/ {,) .(21), 
where A/ is any constant, and 
T(,x') = (1 - {^') . (22). 
T,/(.)=(.*-l)<®(|;)’P„(0.(23), 
P„ denoting the zonal harmonic of degree n. 
In our standard case v is real and greater than unity, and in every case p.' lies 
between the limits + 1 and — 1, and is real. I have adopted the above notation 
(according to which the functions T/ differ in form by the constant factor 
(— 1)^^^) in order to avoid introducing imaginary coefficients unnecessarily.'*^ 
It is easy to see that the solution (21) is applicable in every case. For, if be 
greater than cos^ a, both h and v are purely imaginary ; whilst, if t" be negative, we 
may show that k will be imaginary, but v will be real and less than unity. In any 
case T/ {y) will be either real or purely imaginary, so that (/x) T/ (z^) can be 
always made a real function of the coordinates {x, y, z). Moreover, the right-hand 
side of (21) is finite, single valued, and continuous throughout the liquid spheroid, 
and satisfies the differential equation (6). It therefore only remains to investigate 
the boundary-conditions which must be satisfied by i{j at the surface of the liquid. 
5. The spheroidal harmonics referred to the liquid spheroid, required for these 
boundary-conditions, will be formed as follows ;— 
Let 
P-(?) = ^,(|)'‘(£^+i)"s(-i)»^P„(‘£) ■ . . (2-1), 
c(0 = (r+i)‘"(|f)'p.(0s(-i)*%'(‘£) • ■ ■ (25), 
* The tesseral harmonics may be replaced by the associated functions of the first kind of Heine ’v\’ith- 
ont any change in the formulae, the constant coefficients being supposed included in A/. 
MDCCCLXXXIX.— A. 2 C 
