STREAMS OF POLARIZED LIGHT FROM DIFFERENT SOURCES. 401 



described, the numerical value of /3 being supposed not to lie beyond the limits and 90°; 

 let v be the velocity of propagation, t the time, \ the length of a wave, and put for 

 shortness, 



Y (•* ~ *) = <P CO- 

 Then about the time t, and at no very great distance from a given point, suppose the origin, 

 we may represent the diplacements belonging to the given stream of elliptically polarized 

 light by 



a/ = c cos /3 sin (0 + e), y = c sin /3 cos (<j£> + e). . . . (2). 



If the light be convergent or divergent, c will depend upon cp, but for our present object 

 any variation of c arising from this cause will not enter into account. The value of a, which 

 determines the direction in which x is measured, as well as that of /3, is given by the nature 

 of the polarization. The polarization is right-handed or left-handed according to the sign 

 of j3. As to c and e, the phenomena of optics oblige us to suppose that they are constant, 

 or sensibly constant, for a great number of consecutive undulations, but that they change in 

 an irregular manner a great number of times in the course of one second. The known 

 rapidity of the luminous vibrations allows abundant scope for such a supposition, since c and 

 e may be constant for millions of consecutive undulations, and yet change millions of times in 

 a second. This series of changes, rapid with respect to the duration of impressions on the 

 retina, but slow compared with the periodic changes in the motion of the ethereal particles, 

 is exactly what we might have expected beforehand from a consideration of the circumstances 

 under which light is produced, so far at least as its sources are accessible to us; and thus 

 in this point, as in so many others, the theory of undulations commends itself for its 

 simplicity. 



If c were constant c 2 would be a measure of the intensity, so long as we were only com- 

 paring different streams having the same refrangibility. But since c is liable to the changes 

 just mentioned, if we wish to express ourselves exactly, avoiding conventional abbreviations, 

 we must say that the intensity is measured, not by c~, but by the mean value of c 2 , which may 

 conveniently be represented by nt (c 2 ). 



2. Let us examine now whether it be always possible to resolve the given disturbance 

 into two which, taken separately, would correspond to two elliptically polarized streams of 

 given nature. For the sake of clear ideas, it may be supposed that the azimuths and eccen- 

 tricities of the ellipses belonging to these two streams are given and invariable, while the 

 azimuth and eccentricity of the ellipse belonging to the first stream are given for that stream, 

 but vary from one to another of a set of streams which we wish to consider in succession. 



Let a lt Cj, &c. be for the first, and x 2 , e 2 , &c. be for the second stream of the pair, what 

 w, c, &c. were for the original stream ; and resolve all the displacements along the principal 

 axes of the latter stream. Then, in order that the original disturbance may be equivalent to 

 the pair, we must have, independently of rf>, 



#, cos (ci] - a) - y x sin (a t - a) + x 2 cos (a 2 - a) - jfe sin (a 2 - a) = x' ; 1 

 Wi sin (cti - a) + y l cos (a x - a) + w 2 sin (a 2 - a) + y 2 cos (a s - a) = y. J 



Conceive w lt y x , a? 2 , y i} x', and y' expressed in terms of <p by the formula? (2) and the 



