Reflexion and Refraction. 1 1 1 



transversal, as if the other component did not exist, will furnish 

 a refracted ray and a partial reflected transversal uniradial in its 

 direction; and that the total (or actual) reflected transversal will 

 be the resultant of the two partial ones. 



When 9 3 = fl's, the partial reflected transversals will coin- 

 cide, and their resultant will have a fixed direction, independent 

 of the direction of the incident transversal. The angle of inci- 

 dence at which this takes place is the polarizing angle, and the 

 common value of 3 and 6' 3 is the deviation. If, at the polarizing 

 angle, the partial reflected transversals be equal in magnitude, 

 and opposite in direction, their resultant will vanish, and the 

 reflected ray will disappear. This will happen when the inci- 

 dent transversal is in the plane of the two refracted transversals, 

 and therefore in the intersection of this plane with the incident 

 wave plane ; for, when there is no reflected ray, the incident 

 transversal alone must be equivalent to the two refracted trans- 

 versals. 



Since the reflected transversal can be made to vanish at the 

 polarizing angle, this angle might be found directly by putting 

 the vis viva of the incident ray equal to the sum of the vires viva 

 of the two refracted rays, and by making the incident trans- 

 versal the resultant of the two refracted transversals. Resolv- 

 ing the transversals parallel to the axes of co-ordinates, these 

 conditions would give four equations, from which we could 



formed in the uniradial directions. We need not, therefore, consider any case but 

 that in which the resolution is uniradial throughout. 



The incident transversal being denoted by T\, let T$ be the reflected transversal 

 determined by the rules given in the text ; and let the uniradial components of the 

 former be r\, r'\, while those of the latter are ra, r's. Then will 



Ti 2 = Ti 2 + r'i 2 + 2-nr'i cos (ft - fl'i), 

 ?3 2 .= T 3 2 -f r' 3 2 + 2T 3 T' 3 cos (83 - fl'a) ; 



where the signification of ft, fl'i, 3 , 0' 3 is the same as in the text. The vis viva of 

 one refracted ray is m\ (n 2 - T 3 ? ), and that of the other is m\ (iV - r's 2 ) ; there- 

 fore the vis viva of both refracted rays is 



