204 ' MR. G. H. BRYAN ON THE WAVES ON A ROTATING 
In general F («•) does not change sign when k = — cosec a, hut when k = — oo, 
F (k) has the same sign as — 
Hence, the equation F (k) = 0 must have one real root between each of the follow¬ 
ing values :— 
00 , 1, K^, Ko, K^, . . . K„_s, — 00 . 
Thus, if K/(^) is positive, all the roots of (66) are real. Let us now^ examine what 
haj^pens when K/ (^) vanishes and l^ecomes negative. Poincare proves* that, if 
t,t {V) is divisible by I, the equation 
K/(C) = 0 
has no real root; we must, therefore, have n — s even, so that bf [t) is not divisible 
by I 
When Ky(^) vanishes, the equation 
F(k) = 0 
reduces to 
k: (/c — 1) (1 — sird {K:cosa(l — K®sin®a)~-] = 0, 
of which the roots are 
L ^2’ ■ ■ ■’ ^11—S' 
Since n — s is even, the equation 
D^P4^) = 0 
has not zero for one of its roots. Thus, the roots of the period-equation are all real 
and different. Therefore, when K,/ [Q changes sign and becomes negative, the period- 
equation must, at any rate at first, continue to have all its roots real. If it have a 
pair of complex roots, the ratio of K,/ (Q : cj^ {Q must not only be negative, but 
numerically greater than some Jlnite limit. 
16. The roots of the period-equations for the oscillations that are symmetrical 
about the axis of the spheroid are to be separated in exactly the same way. It will 
be sufficient to state the results here. We suppose k^, k^, are the n values 
of K which make 
P« {v) = Pn {k cos a(l — /F siiF = 0 .... (71). 
Let n be odd. One of the above values, viz., k^{h + i) will be zero, whilst = — k-^, 
= — K^, and so on. Also, the ratio K„ (^) must be jiositive. It will be 
found that the period-equation (64) has one root between each of the following 
values of k ; 
K-2’ • • •> bj + • • ’’ °° • 
* ‘ Acta Mathematica,’ vot 7, p. 826. 
