170 TRANSVERSAL VIBRATIONS : 



Hence, contrary suppositions have been made respecting it. 

 In the theories of Fresnel and of Cauchy, the vibrations are 

 assumed to be perpendicular to the plane of polarization, in 

 those of MacCullagh and Neumann, to be parallel to it. 



Professor Stokes has however arrived at a result, in the 

 dynamical theory of diffraction, which seems to afford the 

 means of deciding between these hypotheses. When a polar- 

 ized ray is diffracted, the plane of vibration of the diffracted 

 ray should differ from that of the incident, the positions of 

 the two planes being connected by a very simple relation. 

 This relation may be deduced in the following elementary 

 manner. 



"When a polarized ray is incident perpendicularly upon a 

 fine grating, the direction of its vibrations is (by the principle 

 of transversal vibrations) in the plane of the grating, when 

 the wave reaches it. Let a denote the angle formed by that 

 direction with the lines of the grating : then, if the ampli- 

 tude of the incident vibration be taken equal to unity, it may 

 be resolved into two, namely, cos a, parallel to the lines of 

 the grating, which will be unaltered by diffraction ; and sin a, 

 perpendicular to them. The second component is to be re- 

 solved again, in the direction of the diffracted ray, and per- 

 pendicular to it, respectively ; and of these the latter portion 

 alone is propagated as light. Its value is sin a cos 9, 

 6 being the angle which the diffracted ray makes with the 

 incident. Hence the two components of the diffracted ray 

 are 



cos a, and sin a cos 9 ; 



and their ratio is equal to the tangent of the angle which 

 the direction of the vibration in the diffracted ray makes 

 with the lines of the grating. Denoting this angle by a', 

 we have therefore, 



tan a' = tan a cos 9. 



