530 MR. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS 



+ cot0[(2 + ic*)sin s 6 - 3^] + 



- 4(2 + K 2 ) sin 2 - s 3 ] ^ + 4(2 + K 2 ) sin 2 + .s 2 ] cot 0^ 



=0. . : : . . . - . .-, (62) 



Now, substituting for dU/d0 from (61) and for U from (60), (58) becomes 



sin 



^/ 1\ . Zfi dw dV 



sc {# . s \ 2 8COS 0~2\ l r c) B dO~ S _d0 



~ " * ' 2 2 



~Bin*0\2c ~smB " sin* (2 + K 2 ) sin 2 Q - 



or, multiplying by sin 0[(2 + /c 2 ) sin 2 6 s 2 ], 



- - s.w f(2 + K 2 ) sin 2 - s 2 ] + * 2 s 1+ ^ sin 0eos Ta = 0. 

 2 L\ / J \ /; / do 



Multiply this by s and add it to (62), thus, 



or 



dw 



/ ft o\ 



(63) 



Also, between (60) and (61), eliminate V, then 



.-in^. (64) 



The equations we have to satisfy are (61), (63), and (64). Writing p instead of cos 6, 

 these become 



