in the Undulatat'y Theory. 163 



where t is the time from the commencement of the disturb- 

 ance, while the velocity of the propagation of the wave is 



expressed by u = — = — (2.) and k = (3.) 



A being the wave-length in the medium. 



In common or unpolarized light the vibrations are in all 

 possible azimuths round x ; hence the coefficients a and /3 

 are wholly arbitrary and independent. 



In plane polarized light we have, on Fresnel's principles, 



a = A cos i /3 = A sin / (4.) 



whence i is the angle formed by the plane of vibration with 



another plane, which he terms the plane of polarization : also 



the squares of the amplitudes expressing the intensities of light, 



a2 + /3^ = A2. 



The formulas in this case represent two rays polarized in 

 planes at right angles. 



If we consider only one ray wholly polarized in either 



plane, it is equivalent to supposing either / = 0, or i = — ' 



or that one of the formulas disappears. (5.) 



M. Cauchy, Professor Maccullagh, Mr. Tovey, and other 

 mathematicians term the plane of polarization that in which 

 the vibration is performed. M. Fresnel uses the same term 

 to signify the plane perpendicular to this. This difference in 

 terms, however, involves consequences which affect the sub- 

 sequent applications of the theory*. But this will not influence 

 our present investigation. 



In the case of elliptical vibrations we have to consider 7iof, 

 as in the other cases, a rectilinear displacement and its re- 

 solved parts, but a curvilinear displacement, which is the re- 

 sult of two virtual rectilinear displacements at right angles to 

 each other, and in a plane perpendicular to the ray, and one 

 of which is retarded behind the other by an interval h. Thus 

 the expression will be 



)j = 2 fa sin (nt—kxyi . , 



? = 2 [/S sin («^-/t.r-i)], ^^-^ 



in which, taking a single term, and substituting, we readily find 

 the equation to the ellipse described by a vibrating molecule, 

 the origin being at the centre, the conjugate axes parallel to 

 the coordinate axes of j/ and z^ and their values being 



« sin 6 = \ axis /3 sin ^ = | conj. axis (7.) 



• See L. & E. Phil. Mag. and Journal of Science, vol. xii. No. 74, p. 259. 



M2 



