528 MB. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS 



The first two of these are 



i 3m n\/&u, ,a^"\ , n dw3m n / , 3m 



+ 



m + n 



&u, ,a^"\ , n dw3m n / , 3m n\ 



5^ + cot 0-^+2^- - 1 + - ~ ) W cot 2 8 



tiffi dOJ 30 m + n \ m + n/ 



, / 3m n\ / 3m_ _n\ cos 30 . 3m w 1 &v 1 3*w _ , 



" m + n)* ~~m + n)sirf6d4> + m + n still 30 ty + tirfB ty* = ' 



i (& , i a dv\ , a a \ 2 3m n dw cos 6 3u / 3m n\ 



+ U^ + cot d .- + v (2 cosec 2 ^) + ^ 3 - - - -f- - -7,^7(2+- -) 



> < 0- 361 sin m + n c<f> sin 2 d< \ m + n ] 



1 / 3m w\3^ 3m n 1 9% 

 rsh?0l/ "l^T^/a^ H '^T+^sTn^S^ = 



Substituting from (48), these are 



2 cos 3w , 



- + -=0, . (51) 



, . 



-^ + = 0, (52) 



and, writing 



, ....... (53) 



m + n 



(48) becomes 



** /^* i A /3 1 d"\ /c ,\ 



-; = c^+ W cot^ + - 1 ^. ...... (54) 



Substicucing for w, we find 



a = 0, (55) 



J f fl _L _- 



H ^ + 



= 0. (56) 



Since u, v, w must be the same for </> + 2n as for ^., we may put 



w a cos s(f>, v oc sin s<^>, w a cos s<, 

 where s is an integer. 



