530 PROFESSOR KELLAND ON THE VIBRATIONS OF 



. . (i+r cos a) cos e k 

 Also cos y = v ; '- z + - by (3 



r o 



r sin a cos (b e I 

 sm y = -- 4 by (4) 



Te 76 J V J 



(i + r cos a) cos p rk 

 cos 7 = > *- + - 



sin <y = 



5 <r r ff 



r sin a cos <p rl 



re re 



Fronl these equations we obtain 



p 2 (e 2 + A 2 )+2p sin <b (hir cos a er sin a) 

 t 2 (e 2 + h 2 ) + 2 t sin $ (hir cos a er sin a) 



or (^ 2 ~P 2 ) (c 2 + A 2 ) + 2 (/ p) sin (hir cos a er sin a) = 



and (,2g2^ (^ + A 2 ) + 2 (r a) cos (ki+r cos 



r (f 4-p) (e 2 + A 2 ) + 2 sin ^> (hir cos a-er sin a) = . . . (!') 

 l(r+ir) (/ 2 + A 2 ) + 2cos (Af+r cos a + ^r sin a) = . . . (2') 

 By substituting the values of cos /3, cos p, &c. given above, in the remaining 

 four equations, they become 



. N (i r cos a) sin (b . . esr 



(a p + b () * r - r.sm a sin <p + - r sin a sin (b 



tg re 



aQth + btOh , erlserl n 



+ ~ ~ -- +tan< -- - = ............ 5 



. , r sin a sin d> ,. , . (a sr) sin d) (i+r cos a) 



(a p + b f) - *- (i + r cos a) sm d> + 5 - '- 



t p 5 



apte + bg(e , erk srek - 



____Ji + tan <p - =Z. = ............. (6) 



r e tQ cos 



aerk+brek , Ot 



sin a cos - a) co. - C S 



_ _ - . 



r <r f p cos <p 



erl+bTgl ,0th stgh A 



- + tan <p * - s = .............. 1 8) 



r ff / p 



From (5) and (6) by eliminating a p + & t we get 



. . , (1 + sr 2ff) .. 2 \ ,*. 

 r 2 sm 2 a sin ^ - + (r r 2 cos 2 a) sm a> 



v 



r 



(A r sin a + e i r cos a) 



r-p 



, (1 *)ffr ,. . , . 



+ tan <p ' (Ir sm akir cos a; = 



* f tr x r 



r a 



