254 PROFESSOR POWELL'S REMARKS ON THE 



by M. Cauchy, and simplified by Mr. Tovey and Mr. Kelland, involves certain 

 conditions, viz. the evanescence of certain terms ; the interpretation of which implies 

 peculiar views of the constitution of the ether. 



Whether a direct and complete solution, without the introduction of these condi- 

 \^tions, can be attained, does not appear. But Mr. Tovey has been the first (as far as 

 I am aware) to point out (in one of his valuable series of papers in the London and 

 Edinburgh Journal of Science) that, without these conditions, a certain form of the 

 wave-function is a particular solution of the equations ; and this form is precisely that 

 expressing elliptically polarized light. 



(4.) Upon a careful examination of that most important and ingenious investiga- 

 tion, it appeared to me that there were one or two questions connected with the sub- 

 ject still requiring to be cleared up. 



If the absence of the conditions in question be essential to the case of elliptically 

 and circularly polarized light, it follows that all the preceding investigations (which 

 depend on the fulfilment of those conditions) are applicable only to unpolarized and 

 plane-polarized light, and consequently the general integration is limited in a most ma- 

 terial part of its application ; a defect which is only remedied by the supplementary 

 investigation of Mr. Tovey, in which, for this case, a particular solution is assigned. 

 It seemed, then, necessary to show explicitly that the non-fulfilment of the condi- 

 tions, that is, the non-evanescence of the terms in question, is essential for elliptically 

 polarized light, as their evanescence is for common light ; and thus to exhibit distinctly 

 the relation between the cases of elliptically polarized, and those of plane-polarized and 

 unpolarized light : and again, to remove, if possible, the obscurity and discrepancy 

 of opinion in which the physical interpretation of those conditions, with regard to 

 the supposed constitution of the etherial medium, appeared to be involved. 



With this object, it seemed necessary to take up the subject from its first principles; 

 and, in pursuing the inquiry, I gladly acknowledge the valuable suggestions I have 

 derived in the course of some correspondence with Mr. Lubbock. I have been thus 

 led to combine the elucidation of the points just referred to, in a connected view 

 with preceding results, by a simplified method ; and to trace clearly the relation be- 

 tween the changes in the conditions supposed with regard to the ether, the resulting 

 state of polarization in the ray, and the expression for the dispersion. 



Analytical Investigation. 



(5.) For the purpose of the ensuing investigation it will be necessary to premise a 

 brief statement of the formulas adopted from the undulatory theory. On the ordinary 

 principles of that theory, the form 



u=^ {(x,^m{nt ^hx)} (1.) 



is the sum of a number of functions, each expressing a single wave, whose aggregate 

 constitutes a ray of light ; (u) is the displacement of the vibrating molecule of ether ; 



