468 Prof. Challis's Explanations of Phenomena of Light 



X 



be an odd multiple of -, the value of (w 2 -f v 2 )* and the resultant 



of the transverse accelerative forces will plainly be the same as in 



the former case ; but for the resulting values of w and <j we shall 



have 



aa 8??i(?/ 2 — x*) . 2tt , ' . 



w— — = — —§ -sm — (tcat—z + c). 



k \ A. ' 



Hence w and a will each be very small, on account of the small 

 ratios of y and x to X. And besides this, the dynamic effect of 

 the undulations in the direction of z must be estimated by the 

 sum of the values of a within a small circular space about the 

 axis. But clearly within such a space the positive and negative 

 values of the right-hand side of the above equation are equal, 

 and the sum is consequently zero. Thus we may conclude that 



when the difference of phase is an odd multiple of -, the direct 



vibrations are wholly inoperative. But experiment has proved 

 that the light perceived in this case is precisely the same as in 

 the other. Hence it necessarily follows that the perception of 

 light is in no degree affected by the direct vibrations,] and that 

 it is entirely due to transverse vibrations. 



This law being established, we may next conclude that the 

 undulations of two oppositely polarized rays produce indepen- 

 dent luminous effects, simply for the reason that their transverse 

 accelerative forces act in rectangular directions. Also since, as 

 is known by experience, the luminous effect of a series of undu- 

 lations is independent of phase, it follows that the combined 

 luminous effect of two oppositely polarized series is independent 

 of difference of phase. Thus the theory explains the experi- 

 mental fact, that oppositely polarized rays having a common path 

 do not interfere, whatever be the difference of their phases. 



It is shown above that the undulations of two oppositely 

 polarized rays into which a primitive ray has been separated 

 produce, when they have a common axis, the same luminous 

 effect on the eye as the undulations of the primitive. Hence 

 as the two rays are equal, the intensity of each polarized ray is 

 just half that of the primitive ray. This law may also be deduced 

 from the principle that intensity is measured by the square of the 

 velocity of vibration of the aether. For, on this principle, if the 

 velocity in the primitive ray be called <j>(r), and r make any 

 angle « with the axis of x, the total intensities in the primitive 

 and the polarized rays are respectively measured by 



S$(<t>{r)yrdudr, j j( <£ (r) cos *) 2 rdctdr, JJ((/>(r)sina) 2 r^r, 



the integrations being taken from «=0 to a=27r, and frornr=0 



