176 On a Dynamical Theory of 



reflected transversals will be the same as the resultant of the 

 refracted transversals. 



Hence, recollecting what has heen proved respecting the 

 moments of the transversals applied at the extremities of the 

 rays, we have the following theorem : 



Supposing the length of each ray, measured from the point 

 of incidence and in the direction of propagation, to be taken 

 proportional to the velocity with which the light is propagated 

 along it, and its transversal to be drawn through the extremity 

 of this length, the incident and reflected transversals having 

 their proper directions, but the refracted transversals having 

 their directions reversed ; if all the transversals so drawn be 

 compounded like forces applied to a rigid body, their resultant 

 will be a couple, lying in a plane parallel to the plane which 

 separates the two media. 



This theorem affords a complete solution of the question 

 of reflexion and refraction.* Expressed analytically it gives 

 five equations, of which four are independent. 



To apply the preceding results to a simple case, suppose 

 the second medium, as well as the first, to be an ordinary one. 

 We have then only one refracted ray, and one refracted trans- 

 versal T Z . 



1. When the incident ray is polarized in the plane of inci- 

 dence, the transversals are all in that plane ; and as they are 

 perpendicular to the rays, and the refracted transversal is the 

 resultant of the other two, we have evidently 



, sin fe - 2 ) sin 



r 2 = 



i I - j-. , 2 I -p . 



sin (^ + * a ; sin (f| + H) 



2. When the incident ray is polarized perpendicularly to 



* The same theorem applies to the other case of reflexion and refraction, when 

 a ray which has entered the crystal emerges from it into an ordinary medium, 

 undergoing double reflexion at the surface where it emerges. In fact, the con- 

 ditions (20) and (21) hold good whether the ordinary medium is the first or the 

 second; and in the latter" case, as well as in the former, it may be shown that 

 the condition (22) is fulfilled, and that the theorem above mentioned is true. 



