174 On a Dynamical Theory of 



phases. Making the latter substitution, and attending to the 

 relations just mentioned, we find 



TI COS QI + T\ COS a'i = rr 2 COS a 2 + /r' 2 COS a' 2 , 



(31) 

 TI cos ]3i + T\ cos )3'i = rr 2 cos j3 2 + // 2 cos /3' 2 . 



In these equations, the angles by whose cosines each trans- 

 versal is multiplied are the angles which a plane, passing 

 through the directions of that transversal and of the corre- 

 sponding ray, makes with the planes of y Q z and X Q z . This 

 is evident with regard to the incident and reflected rays. And 

 if we refer to the diagram in the preceding section, it will also 

 be evident with regard to the refracted rays ; for OQ is perpen- 

 dicular to the transversal r 2 , and to the right line OI 7 , which is 

 the direction of the corresponding ray. 



Taking for the point of incidence, let right lines proceed- 

 ing from it represent the different rays ; and let the length of 

 each ray, measured from in the direction of propagation, be 

 assumed proportional to the velocity with which the light is 

 propagated along it. Through the extremity of each ray con- 

 ceive its transversal to be drawn, and let the transversals so 

 drawn have their moments taken, with respect to the point 0, 

 as if they represented forces applied to a rigid body. The 

 length of the incident or reflected ray being considered as 

 unity, the lengths of the refracted rays (as appears by the 

 last Section) are r and r f respectively. Hence, as each trans- 

 versal is perpendicular to its ray, the moments of the incident 

 and reflected transversals are proportional to TI, r'i, and the 

 moments of the refracted transversals to rr 2 , /r' 2 respectively. 

 The equations (31) therefore signify that when the moments 

 are projected, either upon the plane of y Q s , or upon the plane 

 of #o So, the total projected moments are the same for the two 

 media ; or that, if the transversals themselves be projected on 

 either of these planes, the moments of the projections of the 

 incident and reflected transversals are together equal to the 

 moments of the projections of the refracted transversals. 



