Reflexion and Refraction. 105 



transversals n, T 2 lie between the negative directions of x and 

 y, and the transversal T S between the directions of + x and y. 

 Then if 0i, 2 , 3 be reckoned towards the positive axis of z, so 

 that each angle may be 90 when the corresponding transversal 

 points in the direction of z positive, the equations of the trans- 

 versal TI will be 



_* ^- .JL (14) 



tan t/i cos /i sin i l 



and those of T 3 will be 



ft 4* M 



- = = y (15) 



tan 3 cos /i sin // 



Let 



z + .4# + By = (16) 



be the equation of a plane passing through the directions of TI, 

 T 2 and T 3 . To determine A and B, let the variables be eliminated 

 from this equation by means of (14) and (15) successively, and 

 we shall get the two equations of condition, 



tan 0! - A cos /i - B sin ti = 0. ) 



(17) 

 tan 3 -f A cos / t - B sin /i = ; ) 



which, by addition and subtraction, give 



tan 0! + tan 3 

 2 sin ti 



f-l Q\ 



tan 0i - tan 



A = 



2 cos /i 



substituting, in these values, the expressions (13) for tan 0i, 

 tan 3 , we have 



/ . h cos/2 

 B = tan 2 ( sm/ 2 + - T 



/ Asin/2 



A = tan 2 cost 2 - -- . , 

 \ sin /i - sin' 



whence, by making 



tan *= *. A .. (2) 



