198 
MR. G. H. BRYAN ON THE YfAVES ON A ROTATING 
Now, by (48) 
+ f)] 
— _ irr ^ ^ A _i_ ^ ^ 
- 9. 64o + 0 r 02 / 0 ^ 04 + 1 V 
= - iirpy((„^ + 1 ) Co* 4 ) {(f„== Af + f„‘} 
= — 47rpy (4 + 1) (Co) qi (Co) KJrs- 
= — 47rpy G/c^ cot a cosec^ a . (Co) qi (Co) TA (/«.) 
(52). 
8 . To find V,—Supj)ose that the values of this potential inside and outside the 
spheroid respectively are given by the formulae 
V, = (/r) t: (04/ (Co).(28), 
Vq = {p) u,! (O/m;/ (Co).(29). 
Since Vq are due to a surface distribution of surface density ph, therefore, 
- 4:77pyh = 
pVo" 
[841 
44 
0V4 
_0Nj 
_0N_ 
[ 8 ? 
8CJ 
— B (Co) _ (Co) l 
^./(Co) G^(Co)j 
and, therefore, at the surface, by (30) 
[v] = B,/e2‘“''^c‘^'^T,/®^ (/x) 
= 477 -py C/c^ cot a cosec'^ a . b/ (Co) "^t/ (Co) T4 (/r) 
(53). 
9. Lastly, in the forced oscillations, whatever be the variable conservative bodily 
forces or surface tractions producing them, we know that it is always possible to 
expand the value of [V 3 — 2 ^ 2 /p] over the surface of the spheroid and at all times in a 
series of the form 
[ Y -pm =k:-XTX: "0 •> T.'” M ''“v 
ZwKt 
■ (54). 
where is a constant, and the summations may extend to all possible values of k, 
but only to integral values of n and s. The effect of each term may be considered 
separately. To do this, let us take the case when there is a single term only, fie., take 
[Vs - pojp] = .44 (/x) e-*e"-^^.(55). 
