and the Dynamical Theory of Diffraction. 433 



Stokes's solution is based upon the displacement and rate 

 of displacement of his elastic medium. But in an elastic wave 

 there is not only displacement, but rotation also; and the com- 

 ponents of this rotation must satisfy the equation of continuity. 

 But when a wave is broken up at an orifice, the rotation is 

 left discontinuous by Stokes's solution, and hence it cannot 

 be exact. The equation of propagation of the rotation is the 

 same as that of the displacement, and the two are at right 

 angles to each other, and they are both equally important. 



Hence, on the elastic-solid theory as ivell as the electro- 

 magnetic theory , the true solution of diffraction will depend upon 

 the sums of two similar terms. 



When we take account of both the terms, the fundamental 

 properties by which Stokes attempted to obtain the direc- 

 tion of the motion of these particles vanishes, and that 

 problem is impossible of solution by this means. 



In the general equations of p. 426 let all the disturbances 

 vanish except X' and Y v , so that the electric disturbance 

 is in the direction of X and the magnetic in that of Y. 



In order that the energy coming from the two disturb- 

 ances may be equal, we must have 



V 2 K/3& 2 X'y = V 2 K/36 2 Y"\ 2 

 8tt V2VK/ : 8tt V 8tt ) y 

 or 



The electric displacement at any other point of space will 

 then be 



™ b 2 X'v 



87rC oP 6 



j {(2C + C 2 )p 2 -3CV + 30^6-^-v*)^ 



H' = ^5^3 { -3G 2 ,xz-8G^p}e- i ^-^ds. 



OTTKj Q p 6 r 



For showing the peculiarities of the case, polar coordinates 

 are best. Let 6 be the angle made by p with the axis 

 of z, and cf> the angle around it from the plane XY. Let 

 ©', ©", <3>', <E>", and P', P" be the components of the electric 

 displacement and magnetic induction to increase these angles 

 and in the direction of p respectively. Then we have for 

 the electric displacement, 



