178 On a Dynamical Theory f 



sums in the other, we get the three conditions Mant of the 



V 



(T! COS 0i + /i COS 0'i) COS Zj = T 2 COS 2 COS 2 + r' 2 COS _ 



<ng the 

 T! sin t + /i sin 0^ = r 2 sin 2 + r' 2 sin 0' 2 , ? the 



(TI cos 0i - /i cos 0'i) sin \ = r 2 cos 2 sin i z + r' 2 cos 0' 2 sin i\. 



A fourth condition is supplied by the first of the equations (31), 

 in which equation we have to write 



cos ai = sin 0i cos \, cos a'i = - sin 0'i cos i, 



and to substitute similar expressions for cos a 2 , cos a' 2 . 



The right line OQ is perpendicular to the transversal r 2 and 

 to the ray OT. The cosines of the angles a 2 , /3 2 , y 2 may therefore 

 be found by means of the cosines of the angles which the trans- 

 versal and the ray make with the axes of x ti , y, z . 



The cosines of the angles which the transversal r 2 makes 

 with these axes are respectively 



cos 2 cos 2 , sin 2 , - cos 2 sin 2 . 



As the plane which passes through the ray and the wave- 

 normal 08 is perpendicular to the transversal r 2 , this plane 

 makes with the plane of incidence an angle equal to 90 + 2 or 

 90 - 2 . Let a sphere, having its centre at 0, be intersected in 

 the points 8 , T by the right lines 08, OT, and in the points 

 X Q , YQ, Z by the axes of x , y , z ; and conceive the points T 

 and Yo to be at the same side of the plane x , z , the spherical 

 angle T 8 X being 90 + 2 , and the spherical angle T 8 Z 

 being 90 - 2 . Let E be the angle which the ray makes with 

 the wave-normal. Then, the angles which the ray makes with 

 the axes of co-ordinates being measured by the arcs T X , T Y , 

 T Z , the cosines of these angles respectively are 



sin ^ 2 cos e - sin 2 cos 2 sin e, cos 2 sin e, 



cos iz cos c + sin 2 sin 4 sin t. 

 Hence, as the transversal is at right angles to the ray, we 



