LIQUID SPHEROID OF FINITE ELLIPTICITY. 
191 
If K be greater than unity, r and, therefore, also z' will be real. We may take 
{x, y, z') to be the coordinates of a point corresponding to the point {x, y, z) of the 
liquid. We thus obtain a new region of points derivable from the original region by 
homogeneous strain parallel to 2 or by projection. This region may be called the 
auxiliary region, and the surface formed by points corresponding to points on the 
fluid surface, the auxiliary surface. Our problem thus reduces to that of finding a 
suitable value of xjj satisfying Laplace’s equation (9) within the space bounded by the 
auxiliary surface. 
But we must revert to the original system in order to satisfy the boundary- 
conditions, which must hold at the actual surfaee of the liquid, not at the auxiliary 
surface. If the surface of the liquid be free, y> must be constant over it, and, therefore, 
the condition to be satisfied all over the disturhed surface of the liquid is 
i// = i + y”) + const.(10). 
In forming the expression for V] we must remember that the gravitation potential 
is due to the disturbed configuration of the liquid mass. 
If K be less than unity, t will be imaginary, and, therefore, the auxiliary surface 
will also be imaginary. But the results arrived at by this method in the case 
where r is real will still hold good even if r be imaginary, provided that the expression 
obtained for t/; is a real function of the coordinates x, y, z. The method breaks down 
if K = dh when r vanishes ; this must be treated as a limiting case. 
Solution for the Spheroid hy Spheroidal Harmonics. 
4. Let the liquid be in the form of a Maclaurin’s spheroid the equation of whose 
surface is 
so that 
X- -t y" , _ a;- + _ 
d + 1) d cosec” u'c- cot- a 
Co = cot a . . . 
= 1.(11), 
■ • ( 12 ). 
and sin a is the eccentricity of the spheroid, c being the radius of its focal circle. 
The locus of the corresponding point (x, y, z) is the auxiliary quadric 
xr -h ?/“ 
c- cosec® a c® cot® « 
+ 
T~Z 
— I 
(13). 
This quadric will be a prolate spheroid if lies between zero and cos^ a, that 
IS, if lies between unity and cosec® a. If t® is greater than cos® a, or ac® greater 
than cosec® a, the spheroid will be oblate. If r® be negative, or /c® less than unity. 
