192 
MR. G. H. BRYAN ON THE WAVES ON A ROTATING 
equation (13) represents a hyperboloid of one sheet, but the part corresponding to 
the liquid surface is the imaginary portion for which + 'if' is less than (? cosec~ a, 
and z'^ is negative ; this is the imaginary auxiliary spheroid. 
We shall take as our standard case that in which equation (13) represents a 
prolate auxiliary spheroid. Let it be written in the form 
so that 
Solving for vq, k, we find 
+ ^ 1 
/d(^o^-i) Z.-V 
k^ {vf — 1) = cosec^ a, 
, O „ cot^ « cot® « . /C® 
/C-Pq- = - ^ z= - - 
T" /C" — i 
K COS « 
v/ (1 — /c® sin® a) 
(ij). 
(15), 
cosec® a. — K" 
/c® — 1 
(16). 
The solution of the differential equation (9) must be effected by means of spheroidal 
harmonics applicable to the auxiliary spheroid (14), whilst the expressions for the 
gravitation potential of the liquid mass and the boundary-conditions will involve 
spheroidal harmonics referred to the actual liquid spheroid (11). We must, therefore, 
use tvm different sets of oi-thogonal elliptic coordinates for the auxiliary and the 
actual systems. Let these coordinates be denoted by (/r', v, (f)) (p., (f>) respectively, 
and let them be connected with the rectangular coordinates in the two systems by 
the relations 
X =■ k ^ [p~ 
y = ky/{v^ 
z! — kvy! 
and, therefore, 
Z = kpfJLT 
1) v/ (1 — cos c V (C^ -h 1) v/ (1 
1) \/ (1 — sin (f) = 1) a/ (1 
= cCp/t 
P^) cos (j) 
p^) sin (f) 
V 
(17). 
The surfaces of the spheroids will be given by the equations 
^ = Co .(18), 
or 
V — vq .( 18 a); 
moreover, all over these surfaces, at corresponding points. 
p = p 
(19). 
