Crystalline Reflexion and Refraction. 53 



When 6^-=.e\, the partial reflected transversals will coincide, and their resul- 

 tant will have a fixed direction, independent of the direction of the incident 

 transversal. The angle of incidence at which this takes place is the polarising 

 angle, and the common value of 6^ and 6'^ is the deviation. If, at the polarising 

 angle, the partial reflected transversals be equal in magnitude, and opposite in 

 direction, their resultant will vanish, and tlie reflected ray will disappear. This 

 will happen when the incident 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 transversals. 



Since the reflected transversal can be made to vanish at the polarising angle, 

 this angle might be found directly by putting the vis viva of the incident ray 

 equal to the sum of the vires vivce of the two refracted rays, and by making the 

 incident transversal the resultant of the two refracted transversals. Resolving 

 the transversals parallel to the axes of coordinates, these conditions would give 

 four equations, from which we could eliminate the two ratios of the three trans- 

 versals, together with the angle at which the incident transversal 1^ inclined to 

 the plane of incidence. In the equation produced by this elimination, the angle 

 of incidence would be the polarising angle, and the other quantities would be 

 known functions of that angle; whence the angle Itself would be known. 



It deserves to be remarked, that, at any angle of incidence, if the incident 

 and reflected wave planes be intersected by a plane drawn through the two 

 refracted transversals, the Intersections will be corresponding transversal direc- 

 tions ; that is to say, if the incident transversal coincide with one intersection, 

 the reflected transversal will coincide with the other. For it is evident, from our 

 fourth hypothesis, that if three of the transversals be in one plane, the fourth 

 transversal must be in the same plane. 



of Fresnel ; a very curious fact, which not only shows that the laws of reflexion and the laws of pro- 

 pagation are perfectly adapted to each other, but also indicates that both sets of laws have a common 

 source in other and more intimate laws not yet discovered. Indeed the laws of reflexion are not 

 independent even among themselves ; for the expressions (iii.) and (iv.) in the note on ordinary 

 reflexion (page 43) have been deduced solely from the principle of equivalent vibrations, and^^yet 

 they satisfy the law oi vis viva. Perhaps the next step in physical optics will lead us to those higher 

 and more elementary principles by which the laws of reflexion and the laws of propagation are linked 

 together as parts of the same system. 



