STREAMS OF POLARIZED LIGHT FROM DIFFERENT SOURCES. 403 



Hence, universally, a stream of elliptically polarized light may be resolved into two streams 

 of elliptically polarized light in which the polarizations are of any kind that we please, but 

 different from one another. 



Substituting for q 2 , &c. their values in (7), and replacing y lt y 2 by a x - a, a 2 -a, for 

 which they had been temporarily written, we find 



{ sin (/3 2 - )3) cos (eta - a) - \/- lcoa (fi 2 + /3) sin (ct 2 - a ) \ _1 G, ) 

 = }sin (/3 - ft) cos (a, - a) - v^icos (/3 + /3,) sin ( ai - a)} -" G 2 \ • (8) 

 = | sin (/3 2 - ft) cos (a 2 - a,) - y/ - 1 cos (/3 2 + /3,) sin (a 2 - ai)} -1 G ' 



3. Among these various modes of resolution there is one which possesses several peculiar 

 properties, any one of which might serve to define it. Let us in the first place examine under 

 what circumstances the intensity of the stream made up of the two components is independent 

 of any retardation which the phase of vibration of one component may have undergone rela- 

 tively to the phase of vibration of the other previously to the recomposition. 



For this purpose there is evidently no occasion to consider the manner in which a L , 6, , e, , 

 « 2 , b 2 , e 2 are made up of a, b, e, but we may start with the components. Let p lt p 2 be 

 the retardations of phase which take place before recomposition, and resolve the disturbances 

 along the axes of x, y. We shall have for the resolved parts 



uc = S \a x cos a x sin ((p + e x - p x ) - 6, sin a t cos (<p + ei - p,)} 

 y = S \a x sin e^sin^ + e Y - p,) + fe^oscticos (0 + ei - pi)}, 

 where S denotes the sum of the expression written down and that formed from it by replacing 

 the suffix 1 by 2. To form the expression for the intensity, or rather what would be the in- 

 tensity if the quantities c and e were absolutely constant, not merely, constant for a great 

 number of successive undulations, we must develope the expressions for a? and y so as to contain 

 the sine and cosine of <p + k, and take the sum of the squares of the coefficients, k is here a 

 constant quantity which may be chosen at pleasure, and which it will be convenient to take 

 equal to 6] - pi. If I be the intensity, in the sense above explained, or as it may be called the 

 temporary intensity, we find, putting § for e 2 — p 2 — e x + p x , 



/ = !«! cos a x + a 2 cos a 2 cos $ + b 2 sin a 2 sin S} 3 + { - 6, sin a Y + a 2 cos a 2 sin § — b 2 sin a 2 cos S} 2 

 + {a x sin ei! + a 2 sin a 2 cos 3 - b 2 cos a 2 sin 3} 2 + {b x cos a, + a 2 sin a 2 sin 5 + b 2 cos a 2 cos $J 2 

 = a, e + bf + a/ + H* + 2 (a x a 2 + 6j6 2 ) cos (a 2 - a x ) cos 3 + 2 (a^ + a 2 6,) sin (a 2 - aj) sin S. 

 On putting for a, b their values c cos /3, c sin /3, this expression becomes 

 / - c, 2 + c 2 2 + 2c,c 2 {cos (et 2 - a 2 ) cos (/3 2 - /3,) cos $ + sin (a 2 - a,) sin (/3 2 + /3,) sin 3} . (9). 

 In order that / may be independent of the difference of phase p 2 - p k , and therefore of 3, 

 we must have either 



cos (a 2 - a : ) = 0, sin (/3 2 + ft) - 0, . . . . (10), 



or sin (a 2 - a x ) = 0, cos (ft - ft) = 0, . . . . (n). 



The equations (10) give a 2 - a x = ± 90°, ft = - ft , so that the ellipses described in the 

 case of the two streams are similar, their major axes perpendicular to each other, and the direc- 

 tion of revolution in the one stream contrary to that in the other. It will be easily seen that 

 Vol. IX. Part III. 52 



