468 Mr. Sylvester on the Optical Theory of Crystals. 



By analytical geometry : 



cos \ t = cos CO . cos co t + cos $ . cos <p, + cos \J/ . cos \|/, 



" V (a 2 - F) (a 2 - c») * V a 9 - c 2 



/fa 2 ~- c 9 ) ( y - c 2 ) / c 2 -&* 



V (c 2 - a 2 ) (c 2 - 6 2 ) ' V c* - a 2 



• ( P< «- a 2 ) (V - a*) + • (p» - c 2 ) (V - c 2 ) , 

 (a 2 - c 2 ) 

 and similarly 



cos i yi = cos co . cos co u + cos $ . cos fyj -f cos \J/ . cos ty u 



_ j/(g - fl 2 ) (g,, 8 - « 2 ) - V{vf - c % ) (v n * - c 2 ) 

 a 2 - c* 

 Proposition 9. 

 To find v t v u in terms of c t c ir 

 By the last proposition 



cos , cos i - W-^)K g -^)(^-^K 9 - c2 ) 

 cos i, . cos t tl _ ^ __ 2 



_ (« 4 -c 4 ) -« 2 - c *. P/ 9 + *„' 

 (a 3 - c 2 ) 2 



_ (a* -f c») - tTTQ 

 (a 2 - c 2 



.*. v? + 2^ = a 9 -f c 5 — (a* — c 2 ) cos * y cos i ir 



Again, 



(sin »,) 2 . (sin *„) 2 = 1 — cos i/— cos ij + (cos » y ) 9 (cos i yi ) 2 



fa 2 - a 2 ) (V - a 2 ) + fa 2 - c*) (V - c 2 ) 

 - L * ' ~~ (a 2 - c 2 )* 



, (« 2 4- c 2 ) 2 - 2 (a g + c 2 ) W + V) + fo , 9 + V) 2 

 + ( fl * _ t*f 



(a* - c 2 ) 2 

 .\ t>, 2 — v ; * = (a* — c 9 ) sin i, . sin i u 

 but a, 9 -f v t ? = (a 9 -f c 2 ) — (« 9 — c 9 ) cos h cos i a 



9 a 2 -f c 2 a 2 — c 2 . , x 

 •'• *>/* = — „ 5 cos ('/ + •/,) 



