54-0 Mr. Sylvester's Analytical Development 



Hence v 2 = b 2 — a 1 — c* . cos s t cos e IP 



and in like manner, for the cotyugate line of vibration. 



v t 2 = b 2 — (a 2 — c 2 ) cos ii| . cos e,/. 



Proposition 12. 

 To find e y e /; in terms of *, »,, 

 (cos e,)* + (cos ej 3 



= 2 (cos a) 2 . (cos $,) 2 + (2 cos y) 2 . (cos \J/ y ) 2 



but by Prop. (9) 



v? = a 2 (sin ^~y + c 2 (cos ^ L ^)' 



ft* = «? ( sin -^yr) t c * ( cos ^^T/ 



.-. (cos 6/ ) 3 + (cos g« = ^.^ „-,,;. rin<// 



multiplied by 



2 { W<((co S "+',) 2 ( Si n^) 2 + (co S ^) 2 (-"7" )') } 



(a2-C 2 )5 



6 2 - vf 



(a 2 — c) sin i t . sin 



and we have seen that 



b % - v? 

 cose, cos £// = - r= - -^ 



^{(sin.^ + fsini,,)*} 



//A 8 — **\ sin/, 4- sin i., 



\ COS 5, + COS £// = A / ( -a L \ . ,.,.,_. 



' u V \« 2 — c J - v sin i . sin » /y 



/(ft 2 — v?) sin i — sin <,, 



cos i j - cos s u = a / ^ ^ . . . . JL 



V a 2 - c* ^/ s j n ; . s i„ r 



•(cos g/ )=A/|^4. *A 

 V l «^ — C* sin »„ J 



sin *, 



V L«- — c 



sin ij J* 



