1 78 On a Dynamical Theory of 



sums in the other, we get the three conditions 



(T! COS 0! + r'i COS S\) COS \ = T 2 COS 2 COS l a + r' 2 COS 0' 2 COS / 2 , 



TI sin 0! + T'I sin / 1 = r 2 sin 2 + r' 2 sin 0' 2 , (34) 



(TI cos 0, - r'i cos / i) sin e\ =r 2 cos 2 sin ^ + 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 H! = sin 0! cos i, cos a\ - - sin 9\ cos i i9 



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 a? > y^ *o 



The cosines of the angles which the transversal r 2 makes 

 with these axes are respectively 



cos 2 cos iz, sin 2 , - cos 2 sin i 2 . 



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

 normal OS is perpendicular to the transversal T 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 $ , T Q by the right lines OS, OT, and in the points 

 X 09 Fo, ZQ by the axes of ar , ^o, s ; and conceive the points T Q 

 and YQ to be at the same side of the plane # 0} * > ^ ne spherical 

 angle T S X being 90 + 2 , and the spherical angle T S 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 Q , T Y , 

 T Z , the cosines of these angles respectively are 



sin it cos c - sin 2 cos 4 sin e, cos 2 sin c, 



cos iz cos c + sin 2 sin i z sin t. 

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



