216 
MR. G. H. BRYAN ON THE WAVES ON A ROTATING 
multiples of 'Ll. We then write (f) cot for cf)', and resolve all j^^^oducts of sines and 
cosines involving cj) or t in terms of circular functions of sums and differences. The 
expression Vg will now be a triple series of the same form as in equation (88), and 
the heights of the tides due to each term of the series may be determined separately 
by the application of equations (89), (90). 
26. Suppose that the attracting mass rotates round the spheroid in the equatorial 
plane in a circle of radius c.y/(Z^ + 1) with angular velocity (1 — 2l) oo (referred to 
fixed axes). 
We then have /jli = 0, ^i = Z, (f)\ = — 2l) cot, cf)' — (f)\ = cf) 2lcot. Therefore 
Vg = my/c “(2/1 + 1) [P„ (/^)p« (C) (0) (Z) 
+ 22'^“[(■// — s)!/(n + s)\} T,/*^ (/r) C {Cj T,/*^ (0) ^^,/(Z) cos s ((^ + 2 /&j^)] (101). 
We have 
P«(0) = 0, (?i odd), 
T;/®^ (0) = 0, {n — s odd). 
(-+ s)! 
'' ’ 2'Gi(w + s)!.i(/i - s)!’ 
[n — s even). 
so that the expansion only involves harmonics which are symmetrical with respect to 
the equatorial plane. 
The first term in the above expansion is a harmonic of the first degree and rank, 
and determines the motion of the spheroid as a whole about the centre of mass of the 
spheroid and the attracting body. This motion can be taken separately. 
Taking, in the usual way. 
h = s;:; {aw (/x) + s;:; a/Tv^> {y.) coss{cf> + 2icot)} 
. ( 102 ), 
where the summation includes only even values of 11 — s, we find, by the method of 
§24, 
2n +J. m p„ (0 r„ (0) q„ (Z) 
a = 
o M K„(0 — 4(72 (0/{^ + 1)} 
= (- lf\\{2n-\- 1) 
on 01 ! 
Pn (^1 In (Z) 
M 2« (1 K„ (?) - 4^.3 {X)l{n {n + 1)} 
also, if 'll — s be even, 
(- l)^('^ + *)f (27i + 1) 
, (//even) (103); 
on 
01 — s\ 
M 2".^ {n + s)! (% — s)! 
t/ (?) Un^Z 
I jr w s-x _ 4/5 (/s - 1) g3 (?) D^P„ (Vs) _ 
’ 1 " ' sD^Vn{vs) —{Is— l)sec^ a.VsD^'^^VJvs)l{ls) 
(104), 
