362 Mr. R. Moon on the Definition of Intensity 



may be equally seen in the general case, where the oppositely po- 

 larized rays are in different phases. For, representing the rays 

 by 



a cos a. sin X, "V ,,v 



a since .sin (X-f D), J 



the first may be replaced by two oppositely polarized rays repre- 

 sented respectively by 



a cos 2 a . sin X, a cos cc sin ct . sin X ; . . . (2) 



and the second by two similar rays represented by 



tfsin^&.sin (X + D), — «cos«sina . sin (X + D). . (3) 



Hence, combining the expressions for rays polarized in the same 

 plane, we shall have in place of (1) two waves polarized in oppo- 

 site planes, respectively represented by 



a A (cos 2 ct + sin 2 a cos D) . sin X -f sin 6 a, . sin D . cos X \ , 



a. cos a sin a. |(1 — cosD) sin X— -sin D . cosXj- ; 



which may be written 



as/ cos 2 a. + sin 2 a cos D) 2 + sin 4 a . sin 2 D . sin (X + D X ), 



a . sin a cos a . \/ (1 — cos D) 2 + sin 2 D . sin (X + D 2 ) ; 



or 



0\/cos 4 u + sin 4 a -f 2 sin 2 a cos 2 a cos I) . sin (X + DJ , 



a sin a cos a \/2 (i — cos D) . sin (X + D 2 ) . 



Now we have just as much right to take the sum of the am- 

 plitudes of these two waves for the intensity at any point of the 

 overlapping beams, as we have to take the sum of the amplitudes 

 of the waves represented by (1) for the like purpose. A com- 

 parison of the results thus derivable, however, will show that 

 they are incompatible, and consequently that Mr. Bosanquet's 

 proposed extension of my definition of intensity in the case 

 of oppositely polarized rays cannot be entertained. 



Undoubtedly, however, I may be expected to state how I pro- 

 pose to estimate the collective effect of the oppositely polarized 

 rays in the circumstances referred to ; and this I shall have no 

 difficulty in doing. 



If this collective effect is capable of being expressed by a 

 function of the intensities of the two waves when acting sepa- 

 rately, whatever be the phases of the latter, we shall have 



intensity = F ia cos «), [a sin fit) j- ; 

 and if we can discover the form of F corresponding to any par- 



