STREAMS OF POLARIZED LIGHT FROM DIFFERENT SOURCES. 413 



These equations give 



/' . */{B° + C 2 + D 2 ) ; J= A - y/(B a - + C 2 + Z> ! ) ; , 



. : B , D (19) 



sin 2 R m ; tan 2 a = — . 



These formulae can always be satisfied, and therefore it is always possible to represent the 

 given group by a stream of common light combined with a stream of elliptically polarized light 

 independent of the former. Moreover, there is only one way in which the group can be so 

 represented. For, though the third of equations (19) gives two values for /3' complementary to 

 each other, these values, as before explained, lead only to two ways of expressing the same 

 result. If we choose that value of /3 which is numerically the smaller, then among the different 

 values of a, differing by 90°, which satisfy the fourth equation, we must choose one which 

 gives to cos 2 a' the same sign as C. 



20. Let us now apply the principles and formulae which have just been established to a 

 few examples. And first let us take one of the fundamental experiments by which MM. Arago 

 and Fresnel established the laws of interference of polarized light, or rather an analogous 

 experiment mentioned by Sir John Herschel. The experiment selected is the following. 



Two neighbouring pencils of common light from the same source are made to form fringes 

 of interference. A tourmaline, carefully worked to a uniform thickness, is cut in two, and 

 its halves interposed in the way of the two streams respectively. It is found that when 

 the planes of polarization of the two tourmalines are parallel the fringes are formed perfectly ; 

 but as one of the tourmalines is turned round in azimuth the fringes become fainter, and at last, 

 when the planes of polarization become perpendicular to each other, the fringes disappear. 



Let the planes of polarization of the tourmalines be inclined at an angle a, and let it be 

 required to investigate an expression for the intensity of the fringes. Since common light is 

 equivalent to two independent streams, of equal intensity, polarized in opposite ways, let the 

 original light be represented by two independent streams, having each an intensity equal to 

 unity, polarized in planes respectively parallel and perpendicular to the plane of polarization 

 of the first tourmaline. If c, c be the coefficients of vibration in the two streams respectively, 

 c cos a, c sin a will be the coefficients of the resolved parts in the direction of the vibrations 

 transmitted by the second tourmaline. Hence we shall have, mixing together, two independent 

 streams, in one of which the temporary intensity fluctuates between the limits (e ± c cos a) 2 

 + (c cos a sin a) 2 as the interval of retardation changes in passing from one point to another in 

 the field of view. The temporary intensity of the other stream being (c sin a) 2 , the intensity 

 at different points will fluctuate between the limits 



m (c 2 ± 2c s cos 2 a + c 2 cos 2 a) + lit (c sin a) 2 . 



It is needless to take account of the absorption which takes place even on the pencils which 

 the tourmalines do transmit, because it affects both pencils equally. Since m (c 2 ) = tlt(c' 2 ) = 1, 

 we have for the limits of fluctuation of the intensity 2 ± 2 cos 2 a. When a = these limits 

 become 4 and 0, and the interference is perfect. When a = 90° the limits coalesce, becoming 

 each equal to 2, and there are no fringes. As o increases from to 90°, the superior 



53—2 



