STREAMS OF POLARIZED LIGHT FROM DIFFERENT SOURCES. 405 



nents, and is therefore constant, that is, independent of p^-p^, and is accordingly equal to 

 what it was at first, when p t and p t were each equal to zero, that is, equal to the intensity of the 

 original stream. 



It may be readily proved from the formula? (8) that it is only in the case in which the polar- 

 izations of the two components of the original polarized stream are opposite that the intensity of 

 the original stream, whatever be the nature of its polarization, is equal to the sum of the inten- 

 sities of the component streams. For, changing the sign of \/— l in these formula;, multiplying 

 the resulting equations, member for member, by the equations (8), and observing that if G' be 

 what G becomes, GG' = g* + h? - c 2 , we find 



{sin 2 (/3 2 - /3) cos 2 (a 2 - a) + cos 2 (fa + fa) sin 2 (a 2 - a) } _1 Ci 2 . 

 - {sin 2 (/3 - fa) cos 2 ( ai - a) + cos 2 (/3 + fa) sin 2 (a, - a) } ~W f . . . . (13) 

 = { sin 2 (fa - fa) cos 2 (a 2 - a,) + cos 2 (/3 2 + fa) sin 2 (« 2 - «,) } "V. ' 



In order that tt^c 2 ) may be equal to m(c! 2 ) + m(c/), it is necessary that c 2 be equal to 

 c i 2 + c*, because, whatever fluctuations Ci and c 2 may undergo in a moderate time, such as the 

 tenth part of a second, c t and c 2 are always proportional to c. Hence the sum of the quantities 

 whose reciprocals are the coefficients of c^ and c 2 c must be equal to that whose reciprocal is the 

 coefficient of c 2 . Since this has to be true independently of fa let the quantities sin 2 (fa - fa), 

 &c. be replaced by sines and cosines of multiple arcs, and let our equation be put under the form 



A + B cos 2/3 + Csin 2/3 = 0. 

 Then A, B, C must be separately equal to zero, or 



sin 2 (fa - fa) cos 2 (ct 2 - a,) + cos 2 (/3, + fa) sin 2 (ag - a,) -= 1 ; j 



cos 2/3 2 cos2(a 2 - a) + cos2/3jCOs2 (a, - a) = 0; [ . . . . (14) 



sin 2/3 2 + sin2/3i ■ 0. ' 



Replacing unity in the right-hand member of the first of these equations by 



cos 2 (a 2 - a,) + sin 2 (a 2 - ai), 

 we find 



cos 2 (/3 2 - fa) cos 2 (a 2 - eti) + sin 2 (/3 2 + fa) sin 2 (a 2 - a^ = ; 



whence fa - — j8,, a 2 = eti + 90°, or else fa and /S, differ by 90°, and et 2 = a,, except in the 

 particular case in which fa = ± 45°, when fa = = 45° satisfies the equation independently of a 2 . 

 Hence the streams must be polarized oppositely, a condition which may always be expressed by 



fa = - fa , a, = o ; + 90°, 

 which equations satisfy the second and third of equations (14) independently of a, as it might 

 have been foreseen that they would, since it has been already shewn that the condition (12) is 

 satisfied in the case of oppositely polarized streams. It now appears that it is only in the case 

 of such streams that this is satisfied. 



6. The properties of oppositely polarized pencils which have been proved, render it in 

 a high degree probable that it is a general law that in a doubly refracting medium the two 

 polarized pencils transmitted in a given direction are oppositely polarized. Were this not 



52—2 



