Reflexion and Refraction. 105 



transversals r ly r a lie between the negative directions of x and 

 ?/, and the transversal r 3 between the directions of + x and - y. 

 Then if ft, ft, ft be reckoned towards the positive axis of 2, so 

 that each angle may be 90 when the corresponding transversal 

 points in the direction of 2 positive, the equations of the trans- 

 versal TI will be 



tan ft cos ti sin i! 

 and those of r 3 will be 



_JL_ = __ = _-_, (15) 



tan ft cos ii sin ii 



Let 



2 + Ax + By = (16) 



be the equation of a plane passing through the directions of r 1? 

 r 2 and r 3 . To determine A and I?, 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 ft - A cos i\ B sin ti = 0, 



(17) 

 tan ft + A cos t - B sin 1 1 = ; 



which, by addition and subtraction, give 



_ tan ft + tan ft 



2 sin tl 



tan ft - tan ft < 

 2 cos i, ' 



(18) 



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



tan ft, we have 



cos* 2 



B = tan ft sint 2 + - 



sm 2 *i - sm*i 2 , 



(19) 

 A = tan ft I cos* 2 - .** 2 j; 



whence, by making 



tan K = . 8 A . 2 , (20) 



