470 Prof. Challis's Explanations of Phenomena of Light 



Since, therefore, both the required conditions are satisfied by 

 means of the equations o-j=5Cos 2 ^ and cr 2 =ssin 2 #, it may be 

 concluded that these equations are true. 



From the above results it appears that the intensities of the 



resolved rays are in the ratio of cos 2 6 to sin 2 6, and that the sum 



of their intensities is equal to the intensity of the original ray. 



If # = 45°, the two intensities are equal, and we have also 



s 



The two rays of this second polarization, like those of the first, 

 produce independent luminous effects on the eye, because their 

 vibrations are in planes at right angles to each other. Hence, if 

 their phases be different, the total luminous effect will remain 

 the same. The compound ray is not, however, identical in its 

 properties with a ray of common light, the resulting transverse 

 vibrations not being of the same character, as will be seen by 

 the following argument. Let the plane of second polarization 

 be now the plane of xz } and let the transverse velocity due to 



the vibrations in that plane be fi(j)(x) sin — - (icat—z + c), and 



that due to the vibrations in the plane of yz be 



fi'4>(y) sin—- (/cat—z + d). 



Then, supposing x } y, z to be the coordinates of a given particle 

 of the sether at any time t, we have 



— = fjb(j>(x) sin— (/cat—z + c), -n=p< r 4>{y) sm ~5~ (/cat— z + d). 



In obtaining from these equations the projection of the path of 

 the given particle on the plane of xy } the variations of z may be 

 neglected; and we may also leave out of consideration the changes 

 of x and y in the coefficients //-<£(#) and /d<f>(y) due to the small 

 changes of position of the particle. By integrating, and elimi- 

 nating t from the results, an equation between x and y of the 

 following form will be obtained : — 



This equation shows that if c—d be zero, or any multiple of ^, 



the left-hand side of the equation is a complete square, and the 

 path of the particle is a straight line. It also shows that in 

 general the path is an ellipse, and in the particular cases in 



which c— d = -~j — . X that it is a circle. These theoretical 



