and the Dynamical Theory of Diffraction. 435 



of polarization. Had we only taken the original disturbance 

 as electrical alone, we should have arrived at Stokes's result. 

 Squaring the coefficients of the time-function and adding, 

 we find 



Hence the light is symmetrical around the axis of Z, and 

 varies from 4 in the positive direction to 1 in the plane X Y, 

 and in the negative direction. 



It now remains to connect the arbitrary displacement XV 

 with the intensity of the original wave. 



Let the arbitrary displacement XW 6W , with the corre- 

 sponding magnetic quantity, exist throughout the plane X Y. 

 From considerations of symmetry, the electric displacement 

 throughout space must be parallel to the axis X; and so we 

 can write 



F'= fj^rp {(2C + C 2 y-3C 2 ^ + 3C lF }e-^- v ^, 



z = constant, 



x = — r cos 0, els = rdfyclr = pdfalp, 



y=—r sin (f>, p 2 — z 2 + r 2 . 



The general and exact integral is 



For positive values of z this gives, between the limits p = co 

 and p = z, 



3ibX'v _ yt) 



2 



But for z negative it is zero. Hence such an arbitrary dis- 

 turbance produces a wave in the positive direction, but none 

 in the negative direction. 



No approximation has been made in obtaining this quantity, 

 and it evidently applies to a plane of any size, even infinitesi- 

 mal, provided the point under consideration is infinitely near 

 to it. The displacement near the surface therefore differs in 

 phase \tt from the arbitrary disturbance, but is dependent 

 upon its value at that particular point. 



Hence we can replace any particular wave-surface by a sur- 

 face of arbitrary disturbance whose phase differs \iz from that 

 of the wave. Such a surface of arbitrary disturbance then 



-ib(p-Vt) 



