208 ' MR. G. H. BRYAN ON THE WAVES ON A ROTATING 
Now, whatever the value of a maybe, ^ is always a root of this equation. For, 
if we put K = ^ in the left-hand side, it becomes 
- i {(^' + (0 (0 V (0 - ^2' (0 V (O/ri 
(f3 + 1) J ^2 + 1 
ciK 
^ 0 
(77), 
as was to be proved. 
Substituting from the relation just found in (76), the period-equation becomes 
(4k:® — d/c® -h k) taiF a — {2k — l) sec® a = 0, 
or, dividing throughout by (2k: — l)tan®a, the other two roots are given by 
2k:® —■ k — cosec® a = 0, 
whence 
X = =b v/(l + 8 cosec® a)} 
-=i{l± v/(9 + 8H] .(78). 
The expression for is proportional to 
that is, to 
(1 — p,'®)* p'(^^ ~ 1)'^ (^ + 2o}Kt — e). 
z [x sin {2coKt — e) -h y cos {2o}Kt — e)}, 
while the height of the displacement of the surface is proportional to 
trr « {a: sin {2o}Kt — e) y cos {2o)Kt — e)}. 
Ptememl:)ering that this displacement is so small that its square may be neglected, 
it can be readily shown by the usual methods of analytical geometry that, if, as is here 
suiiposed, the ellipticity of the spheroid be finite, the displaced surface is a spheroid 
of the same form and dimensions as the original sphei’oid, and can be obtained by 
turning the latter through a small angle about the line 
X sin {2o)Kt — y cos {2a)Kt — e) = 0, z = 0. 
This will, however, no longer be true if the ellipticity of the spheroid is a small 
quantity comparable with the height of the small displacement, or the surface is 
spherical or nearly spherical. In such cases it will be found that the displaced surface 
