IO4 On the Laws of Crystalline 



sin (i, + la) tan 61 - sin (/, - 1 2 ) tan 3 = sin 2<i tan 2 . (8 



Subtracting from (7) the identity 



sin 2 (d + 1 2 ) - sin 2 (ij, - / 2 ) = sin 2tj sin 2< 2 , 

 tliere remains 



sin 2 (M + /a) tan 2 0i - sin 2 (^ - < 2 ) tan 2 3 



sin 2 2t! / i a sin 2/ 2 \ 



= TTT ---- =17- cos * 5 9 ) 



cos* 02 \i| sm 2<! y 



and this, by making 



mz sin 2< 2 + 2/i sin 2 2 , 



w/i sin 2! 



becomes 



sin 2 (<! + / 2 ) tan 2 0i- sin 2 (ti -i 2 ) tan 2 3 = sin 2^ (sin 2< 2 + 2A) tan 2 2 , (11) 

 which is divisible by equation (8), the quotient being 



sin (<! + t 2 J tan 0! + sin (ii - i 2 ) tan 3 = (sin 2/ 2 + 2A) tan 2 . (12) 



Then, by adding and subtracting equations (8) and (12) we 

 obtain 



^ tan 2 



cos 



h tan 



2 ) tan 2 + -. 



-. -. -- -. 



sm (t, - 8 ) J 



These equations give the positions of the incident and reflected 

 transversals when h is known. 



Now let the. directions in which the transversals have been 

 resolved in equations (2), (3), (4), be taken for the axes of z,x, y 

 respectively ; so that, the origin being at 0, the plane of ccy may 

 be the plane of incidence, and the axis of x may lie in the sur- 

 face of the crystal. And, the reflected ray being conceived to lie 

 within the angle made by the positive directions of x and y, let 

 the initial condition that we have assumed for the angles 15 2 , 

 3 be satisfied by supposing that, when these angles begin, the 



