44 PROFESSOR KELLAND ON THE POLARIZATION OF LIGHT 



The result is 



e dx 



+&c. (6.) 



y j 



This equation, although apparently long and complicated, can, by reason of 

 its symmetry, be reduced to a very simple form. 



The reduction is effected by transforming the co-ordinates of each particular 

 part in such a way, that whilst the axis of z always remains unchanged, the other 

 axes shall vary in such a manner that one of them shall be in the direction in 

 which that elementary motion to which it is due is transmitted. Thus all the 

 portions which Involve a, 5/3, 8 7 will be reduced by changing the co-ordinates 

 to others, one of which is in the direction 10 of incidence, and the other perpendi- 

 cular to it, in the plane of incidence. Again, all the portions which involve d T will 

 be reduced by changing the co-ordinates to others, one of which is in the direction 

 TO of transmission of this vibration, and the other perpendicular to it, and so on. 

 The effects of this transformation will be twofold ; 1, A considerable portion of 

 the expression will vanish altogether by virtue of the symbol 2 ; 2, Those parts 

 which remain will, by virtue of the hypothesis that all the vibrations occupy the 

 same time, be reduced to known forms ; or at least the major part of them. 



For incident vibrations, let i and p stand for the co-ordinates reckoned in 

 and perpendicular to the direction of incidence : then 



dx=i cos (f)+p sm<p, y = isin(f>+p cosff). 



Now, the principle on which the reduction is carried on is this ; after the co- 

 ordinates have been transformed, the values of the expressions are determined by 

 means of the law that in a complete medium extending both above and below the 



d 3 a 

 point under consideration, the ratio of -T-J- to a is e 2 . 



But this ratio is evidently the double of the integral 22 (<p r + ^ p 2 ) sin 2 -^ 



T Z 



<? d)' r ki 



whatever direction i may indicate, or ^ = 2 2 (d> r + > p*~) sin 2 -^ . (7.) 



2 Y 2i 



But further, since r^p 2 + # + 2 , it follows that ^ = 2 2 (0 r + ^ ^) sin 2 y ; and 



also, if Ave suppose the law of force to be that of Newton, a supposition which has 

 been made by me, and I think established on good grounds in several preceding 

 memoirs, we shall have the following relations : 



2 



k i i*-p* ki 



T = S2 ,-sm* - 



