1885.] of some experiments of Frohlich. 255 



diffracted and regularly reflected light measured from the plane of 

 diffraction, 8 the angle between them, and p a constant for any 

 given angle of incidence. 



Rethy obtains this result by taking as the solution of the equa- 

 tions of motion in the medium due to the disturbance at one 

 point of the grating, the values for the displacement of any point 

 given by 



d§> d<$ A 



U = -y- V = r- W = 0, 



dy dx 



where <I> = — cos 2tt \ - - Tn + 8 



r [\ 1 



r being the distance between the two points in question. Now 

 this solution corresponds to the motion which would ensue if each 

 element of the grating, considered as a small sphere, were made to 

 twist backwards and forwards in periodic time T about an axis, the 

 axis of z, the same for all elements, and it is difficult to see how 

 this motion could arise. 



Rethy also considers another possible motion given by 



_ _(f^_ _ dM^ _dM? d?<& 



dzdx' dydz' da? dy 2 ' 



which would arise from the action of a periodic force acting paral- 

 lel to the axis of z at each point of the grating, but according to 

 him this solution does not lead to the equations given above. It is 

 difficult to imagine how the state of things over the grating can be 

 that required for the first solution, and in any case the effects pro- 

 duced by a train of waves passing over a given element of space 

 differ from those produced by causing the particles of the element 

 to perform small vibrations under the action of a periodic force. 

 In the second case the motion is symmetrical round the direction 

 of motion of the particles, in the first case it is not. The two are 

 dealt with in Prof. Stokes' paper on diffraction, sections 31 and 27. 



My object in the present paper is to show how the formula 

 (1) employed by Rethy may be deduced on a certain simple as- 

 sumption from Prof. Stokes' results. 



The assumption is that each particle of the bounding surface is 

 performing small oscillations parallel to some fixed direction de- 

 pending on the polarization of the incident light and the angle of 

 incidence. 



Consider any point on the bounding surface and let a line 

 drawn through this point parallel to the direction of vibration meet 

 a unit sphere in Z (Fig. 1). Let a ray regularly reflected from 

 meet the sphere in R and a diffracted ray in R. Then according 

 to Prof. Stokes the directions of vibration in the rays E , R, so far 



18—2 



