400 PROFESSOR STOKES, ON THE COMPOSITION AND RESOLUTION OF 



expressions. The combination of several independent polarized streams is next considered, 

 and with respect to this subject a proposition is proved which may be regarded as the capital 

 theorem of the paper. It is as follows. 



When any number of independent polarized streams, of given refrangibility, are mixed 

 together, the nature of the mixture is completely determined by the values of four constants, 

 which are certain functions of the intensities of the streams, and of the azimuths and eccentri- 

 cities of the ellipses by which they are respectively characterized ; so that any two groups of 

 polarized streams which furnish the same values for each of these four constants are optically 

 equivalent. 



It is a simple consequence of this theorem, that any group of polarized streams is 

 equivalent to a stream of common light combined with a stream of elliptically polarized light 

 from a different source. The intensities of these two streams, as well as the azimuth and 

 eccentricity of the ellipse which characterizes the latter, are determined by certain formulae, 

 which will be found in their place. 



The general principles established in this paper bear on two questions of physical in- 

 terest. Strong reasons are adduced in favour of the universality of the law, that the two 

 polarized pencils which a doubly refracting medium of any nature is capable of propagating 

 independently in a given direction are polarized oppositely. In strictness, we ought to speak 

 of two series of waves rather than two pencils ; for it is the fronts of the waves, not the 

 rays, which are supposed to have a common direction. The other point alluded to relates to 

 the distinction between common, and elliptically polarized light. It is shewn that the changes 

 which are continually taking place in the mode of vibration may be of any nature, and that 

 there is no occasion, in the case of common light, to suppose the transition from a series of 

 vibrations of one kind to a series of another kind to be abrupt. 



At the end of the paper the general formulae are applied to the case of some actual 

 experiments, but these applications are not of sufficient importance to deserve separate mention. 



1. Consider a stream of light polarized in the most general way, that is, elliptically polarized, 

 and propagated through the free ether. Let the medium be referred to the rectangular 

 axes of a. y, %, the axis of « being measured in the direction of propagation. Let o and 

 a + 90° be the azimuths of the principal planes, that is, the planes of maximum and minimum 

 polarization, azimuths being measured about the axis of * from w towards y. Let the 

 rectangular components of the displacements of the ether be represented by lines drawn in 

 the planes of polarization of the plane-polarized streams which these components, taken 

 separately, would constitute. I make this assumption to avoid entering into the question 

 whether the vibrations of plane-polarized light are parallel or perpendicular to the plane of 

 polarization. If we adopt the former theory, the actual lines in the figures which we are 

 to suppose drawn will represent in magnitude and direction the ethereal displacements ; if we 

 adopt the latter, the same will still be the case if we first suppose all our figures turned 

 round the axis of z, in a given direction, through 90°. 



Let the co-ordinates a/, y be measured in the principal planes whose azimuths are a, 

 a + 90°; let /3 be the angle whose tangent is equal to the ratio of the axis of the ellipse 



