194 
MR. G. H. BRYAN^ ON THE WAVES ON A ROTATING 
and let 
<ln iC) = Pu U) ■ .. 
w/ {0 = t,t (C) (^3 + 1) (^)}2 .(2^)- 
Then the expressions 
(^) (Qjt^s .(28), 
'^0 = {[x) u,f {C)fu,f (Co) .(29), 
are solutions of Laplace’s equation which are finite and continuous, the former 
throughout the interior of the spheroid (C = Co)? fh® latter throughout all space 
outside the spheroid, and vanishing at infinity; while at the surface (C == Co) hoth 
expressions become equal to 
[u] = (ja) .(30). 
The Elevation of the Waves on the Surface. 
6. Let h be the normal displacement at any jooint of the liquid surface, i.e., the 
height of the wave above the level of the undisturbed spheroid. 
Let CT be the central perpendicular on the tangent plane, and (iN an element of the 
outward drawn normal to the surface of the liquid spheroid (11). We readily find 
dx _ 0 // _ Cy hi;_ 
0~c “ c- + 1 ’ 0C ~ + 1 ’ r 
(31). 
and at the surface, since the element dN is a tangent to the curve /x = const., 
(f) = const. ; therefore, 
whence. 
the differentials in (32) being total 
Also 
Vdr)?)- 
so that 
7T dN = c"Co dZ .(33). 
