IO4 On the Laws of Crystalline 



sin (j + a ) tan 0! - sin (^ - * 2 ) tan 3 = sin 2*i tan 2 . (8 



Subtracting from (7) the identity 



sin 2 (i! + ia) - sin 2 (/ t - 1 2 ) = sin 2ii sin 2/ 2 , 

 there remains 



sin 2 (i! + i a ) tan 2 0! - sin 2 (^ - i 8 ) tan 2 3 



sin 2 2i! / w a sin 2i 2 - \ 

 = TTT - -^-x- cos 2 02 ; (9) 

 cos 2 02 \m l sin 2ix V 



and this, by making 



m 2 sin 2< 2 + 2/i sin 2 2 



sinSti 



(10 



becomes 



sin 2 (i! + 1 2 ) tan 2 0! - sin 2 (ti - 1 2 ) tan 2 3 = sin 2^ (sin 2* 2 + 2k) tan 2 2 , (11) 



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



sin (i! + 2 J tan 0i + sin (^ - i a ) tan 3 = (sin 2/ 2 + 2k) tan 2 . (12) 



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

 obtain 



, /> A tan 2 ""i 



tan 0! = cos (i!- ta) tan 2 + -, . , 



+ > (13) 



tan 3 = - cos (i ! + i z ) tan 2 + -. -. 2 . 



sm (/! - / 2 ) 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 ,#, y 

 respectively ; so that, the origin being at 0, fhe plane of xy 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 0i, 2 , 

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



