200 
MR. G. H. BRYAN ON THE WAVES ON A ROTATING 
Expanding P„ (pq) and DP„ (j^qV^o powers of V(^, substituting for Vq by means of (1 5), 
and multiplying the resulting equation throughout by (1 — siiP we find 
4^3 (cot a) I {k cos a)” — ^ cos a)"~“ (l — siiP a) + • • • | 
— nK;i (cot a) I (k cos (9n^ — 1)^ a)”~^ (l — siiP a) + . . . | 
= 0 . . (62). 
This is a rational algebraic equation in k of the degree, involving only even 
powers of k. It is, therefore, satisfied by n values of k occurring in pairs corre¬ 
sponding to values of k^. 
II. Let 5 = 0 , but let n be odd. Then DP,i(i/Q) is not divisible by Pq. Hence, we 
mast multiply the equation (60) throughout by DP,^ (pq) and obtain 
K„ (cot a) DP„ {pq) — 4(22 (cot a.){l — siiP ex) p^ (pq) = 0 . . (63). 
If this be developed in the same manner as in the preceding case, we shall obtain 
422 (cot a)|(K: cos ^ {« cos a)"”^ (l ~ /c' siiPa) -|- . . . | 
— (cot ( 3 t) I [k cos a)"“^ — ^^^9 1 )"'^ a)'^“^ (1 — /c" sin~ a) . | 
= 0 . . (64). 
This is satisfied by ?(--{- 1 values of k, but, as before, the positive and negative roots 
are numerically equal, so that there will only be -Kw + 1 ) different values of /cv 
III. Let s be difterent from zero. Multiplying by the expression 
sD^P„ (pq) -f sec^ ex.{K — l )/k . i^o), 
we find 
sec" a K/ (cot a) (k — l)/'^ • (^o) 
•— [422 (cot a) k{k — 1) — 6'K,/ (cot a)} D®P„ (pq) = 0 
which reduces to the following equation in k — 
{ 422 (cot ex') k{k — 1) — sK,/ (cot a)} I (k cos a)" 
■-2 . {2n - 1)-HI - K- sim a) -f 
— {n — 6’) sec ex . K,/ (cot a) (/c — 1) j (/< cos a)"“*“^ 
(n — s — l)(n — s — 2) , , 
-2 _ ^2n. - 1)- ('^ ' (1 — Sin- a) -f 
= 0 
(65), 
( 66 ). 
