1 82 On a Dynamical Theory of 



(ri cos 0i + r'i cos 0^) cos e'i ~ r 2 cos 2 cos 4, 

 TI sin 0! + r'i sin 0\ = r 2 sin 2 , 



(TI cos 0i - r'i cos 0^) sin e\ = r 2 cos 2 sin e,, (43) 



(ri sin 0i - T'I sin 0\) sin ^ cos ^ 



= r 2 (sin 2 sin 2 cos e a + sin 2 ' 2 tan e). 



In this case, the three transversals are in the same plane, the 

 refracted transversal being the resultant of the other two. 

 Therefore if we find this plane, everything will be determined. 



The axes of # , y w Z Q make, with the incident transversal, 

 angles whose cosines are 



cos 0i cos f'i, sin X , - cos 0! sin *!, 

 and, with the reflected transversal, angles whose cosines are 

 cos 0^ cos i, sin 0\ 9 cos 9\ sin ^ ; 



therefore, by Lemma I., the cosines of the angles which these 

 axes make with a right line perpendicular to the plane of the 

 transversals are proportional to the quantities 



sin (0i + 0'i) sin \, - cos : cos 9\ sin 2*i, sin (B\ - 0i) cos i^ 



Now from the product of the first and second of the equations 

 (43), combined with the product of the third and fourth, we 

 find, by the help of the relations (37), 



2ri/i sin (0! + 0^1) sin e\ = r 2 2 tan \ cos 2 {sin 2 cos i z 



- s 2 (sin 2 cos iz + sin 2 tan e) } . 



From the squares of the first and third of those equations we 

 find 



- Sri/i cos 0i cos 0' t sin 2&\ = r 2 2 tan ^ cos 2 2 (s 2 - 1), 



and from the product of the first and fourth, combined with the 

 product of the second and third, 



2ri r'i sin (0'i - : ) cos \ = T 2 2 tan ^ cos 3 {- sin 2 sin i z 



+ s* (sin 2 sin i 2 - cos / 2 tan c ) J . 



