STREAMS OF POLARIZED LIGHT FROM DIFFERENT SOURCES. 407 



The same views lead to the conclusion that the two pencils transmitted through magnetized 

 glass in a direction oblique to the lines of magnetic force are oppositely polarized. Some theo- 

 retical investigations in which I was engaged some time ago led me to the result that these 

 polarized streams are circularly polarized, as well as those transmitted along the line of magnetic 

 force, and that the difference between the wave-velocities varies as the cosine of the inclination 

 of the wave-normal to the line of magnetic force. I hope at some future time to bring these 

 researches before the notice of this Society. 



8. After these preliminary investigations respecting the nature of opposite polarization, 

 which indeed contain, it is probable, but little that has not already occurred to persons who have 

 studied the subject, it is time to come to the more immediate object of this paper, which relates 

 to the combination of independent streams. But first it will be convenient to state explicitly a 

 principle which is generally recognized. 



When any number of polarized streams from different sources mix together, after having 

 been variously modified by reflexion, refraction, transmission through doubly refracting media, 

 tourmalines, &c, the intensity of the mixture is equal to the sum of the intensities due to the 

 separate streams. 



The reason of this law may be easily seen. The components whereby the disturbance due 

 to any one stream is originally expressed have to be resolved, their components resolved again, 

 and so on ; and of these partial disturbances the phases of vibration have to be altered by 

 quantities independent of the time, and the coefficients in some cases diminished in given ratios, 

 and in some cases suppressed altogether. Each stream has to be treated in a similar way. 

 The final disturbance being resolved in any two rectangular directions, each component must 

 be put under the form U cos (p + V sin (p, and the sum of the squares of U and V must be 

 taken to form the expression for the temporary intensity. All the quantities such as U and V 

 will evidently be linear functions of ccose, csine, c'cose', c'sine', &c, where c, c ... and e, 

 e'... refer to the different streams, so that U for instance will be of the form 



Ac cos e + Be sin e + A'c cos e' + S c sin e' + ... 



where A, B, A', B" ... are independent of the time. The temporary intensity will involve IP, 

 but the actual intensity will involve fll(CP), or 



Xtt*2(A ccose + jBcsin e) 2 + 21tt2 {(Ac cose + Be sin e) (.4' c'cose' + Z?Ysine')}. 



Now the products such as cos e cos e', cos e sin e', &c. will have a mean value zero, since the 

 changes in e and those in e have no relation to each other, and therefore the expression for 

 m (U") becomes 



m2(^ccose + JScsin e) 2 , or 2t!l (Ac cose + 5c sin e)", 



that is, the sum of the quantities by which it would be expressed were the different streams 

 taken separately. 



Two streams which come from different sources, or which, though in strictness they come 

 from the same source, are such that the changes of epoch and intensity in the one have no rela- 

 tion to the changes of epoch and intensity in the other, may be called independent. 



