88 Proceedings of the British Association. 



each other. It is on this supposition alone that all the principal inves- 

 tigations proceed, from which the theory of dispersion is derived. And 

 in all these investigations we consider a rectilinear displacement or 

 vibration, which may be generally in any direction, and whose resolved 

 parts in the direction of the three axes are £ rj £ respectively. This may 

 apply to all cases of unpolarized or plane-polarized light. But for ellip- 

 tically (including circularly) polarized light, it is necessary to consider, 

 not a rectilinear, but a curvilinear displacement or vibration, which is 

 the result of two virtual rectilinear displacements acting at right angles 

 to each other, and in a plane transverse to the direction of the ray, and 

 one always in a phase retarded behind the other by an interval (6). In 

 this case, therefore, it is necessary to proceed by making one of the co- 

 ordinate axes (as x) coincide with the ray, and £=o, A£=o, &c, while 

 the other two in y and z coincide with the components, which give the 

 elliptic vibration, and are of the forms — 



t? = 2 [a sin (nt—kx ] 



I = 2 [|3 sin (nt—kx +&)] 

 This case, I believe, was first considered by Mr. Tovey. Pursuing the 

 investigation thus, taking the axes generally as in any direction what- 

 ever, with respect to the arrangement of the molecules, it appears from 

 Mr. Tovey 's paper, (' Journal of Science,' No. 71,) and from the somewhat 

 simplified form in mine (' Phil. Trans.' 1838, part 2,) that in the case of 

 elliptic polarization, the condition (B) cannot hold good ; while for com- 

 mon or plane polarized light it must hold good. The distinction there- 

 fore between the different states of light as to polarization, depends on 

 this characteristic or criterion, which I call (E). The discussion between 

 Mr. Tovey and Mr. Lubbock (' L. & E. Phil. Mag.' Dec. 1837, Jan. 1838,) 

 seems to turn upon these propositions :— 1. That every system of mole- 

 cules (constituted as supposed in all these investigations) has at every 

 point three axes of elasticity, whatever be the peculiar arrangement of 

 the molecules. 2. That if we take these axes for the axes of co-ordi- 

 nates, then the equations of motion are reduced to the form (C), or the 

 condition (B) holds good. 3. This form of the equation is necessary for 

 the investigation of the wave-surface : or at least, so much so, that 

 without it the deduction is immensely complicated. At all events, the 

 universal existence of such axes is essential to the nature of the wave- 

 surface. Now, since these considerations are essential to the application 

 of the theory to all media, it follows that in all cases there are certain 

 axes in reference to which the condition (B) holds good. This, then, 

 appears at direct variance with the distinction established above, or the 

 criterion (E). And if we set out with equations (C), and pursue a train 

 of deduction similar to Mr. Tovey's or mine, we find corresponding for- 

 mulae, but from which the conclusions in question cannot be derived. 

 It appears, then, essentially important, that this discrepancy should be 

 cleared up, and the fallacy, if any, detected. 



