528 PROFESSOR KELLAND ON THE VIBRATIONS OF 



Our equations become 



(ir cos a) sin (^ (tg] cos /?; r sin a sin <p = (tg} sin /3 



(? + r cos a) cos (j) = (rg) cos 7 ; r sin a cos (f) = (r) sin 7 



r sin a sin (f> = (a p + 1> t) cos {3 + (sst) tan (p sin 7 



(' + / cos a) sin <=( g + bf) sin /?+(** r) tan ( cos 7 



r sin a cos = (a + 6 r) cos 7 (p * tan < sin /3 



(i r cos a) cos 0= (a ff+6 r) sin 7+ (p * f) tan < cos /3 



By substituting in the last four equations the values of cos /3, cos 7, &c, from 

 the first four, we obtain 



ap + b t ,. s r 



r sin a = - (t r cos a) + - - r sin a 



t re 



, a p + b t . esr ,. 



ft + r cos a) = * - r sin a + - (i + r cos a) 

 g re ^ 



a e+b r,. . p / , 



r sin a = -- u + r cos a) + -S - tan 2 9 r sin a 

 r <r v r p 



a g+br . p s t . . . 



t rcos a = - r Bin O + * - tan-* m (ir cos a) 

 rg tg ' 



I <ssr\ a + bt , 



Hence I 1 -- I r sin a = - - (ir cos a) 



\ 5 a J t p 



/ .. gsr\ .. s ap + 



[ 1 -- J u + r cos a) = j- 



V r-tf / f- 



ap + bt 



> sin a 



i 2 -*- 2 cos 2 a=r 2 sin 2 a 

 or '= 



Moreover, by means of the first four equations we get 



(i 2 r 2 ) sin cos (> = (( p) (r ) cos (fl 1 



7= 4 j 



a. If ^?,a=0 

 6. If =, a=?r 



7T 



c. If p 7= -y sin p=cos 7, cos /3= sin7 



and 



sina 



= tan /3 = cot 7 = ; 



1 cos a 

 7 = | and /3 = ^ + ~ 



sin a 



