Electromagnetic Theory of Light. 237 



therefore e/Se/ur = (m/j,~ l c + Sct/lkt) —1 <r, 



whence, operating by Sfter( ), 



1 = Sff/*(mft —, C + Scry^O") -1 <7, 



aud this may be put in a variety of forms, as usual. But the 

 interesting thing to notice is that by the above method it is 

 obvious that by a similar process the equation of the wave- 

 surface is 



and that this has been arrived at without any complex inte- 

 gration of (9) and (10). 



In concluding this part of the paper I may remark that the 

 above methods can be applied to other theories of light. 

 They are not quite so naturally applicable because in such 

 theories — without exception as far as I know — only one of 

 the pairs of vectors 8, e and /3, <y obviously presents itself, 

 and the other pair must be deliberately introduced by 

 definition. In the electromagnetic theory both pairs are 

 present before the special problems connected with optics are 

 considered. 



II. Reflexion and Refraction at the Surface of Crystals, treated 

 by a Theorem of Sir Win. Rowan Hamilton. 



In his i Elements of Quaternions' (423 (12) in 1st edition, 

 probably this important theorem will not escape the editor of 

 the 2nd edition) Hamilton has proved a theorem of such 

 singular beauty and ease of application in particular cases, 

 that I think it well worth calling attention to. This seems 

 the more necessary as it appears to have been overlooked by 

 subsequent writers. 



His treatment is — probably necessarily, based as it is on 

 MacCullagh's theory — somewhat complex, whereas the treat- 

 ment on the electromagnetic theory is little more than an 

 interpretation of the fundamental equations. He treats only 

 of the case where a ray in an isotropic medium is incident 

 on the face of a crystal, and where the polarization of the 

 incident light is such that there is but one refracted ray. On 

 the electromagnetic theory it is just as easy to suppose both 

 media to be crystalline, and to allow the incident polarization 

 to be arbitrary. 



There will in this case be one incident ray, two refracted 

 rays, and two reflected rays, five in all as against Hamilton's 

 (and MacCullagh's, since in this particular Hamilton is 

 following MacCullagh) three. 



Let the M.M.F.'s of the five rays at the point of incidence 



