Electromagnetic Theory of Light. 229 



magnetic wave-surface. The present writer (Phil. Mag. xlii. 

 Sept. 1896, p. 224) pointed out that it is an orthogonal pro- 

 jection — in two different ways — of the Fresnel surface. I 

 do not know of other writers who have contributed to this 

 particular department ; but I suspect that this is due to nvy 

 ignorance, and not to the absence of literature. 



The following statements are either explicitly made in, 

 or are obviously deducible from, the text of Basset's ' Physical 

 Optics.' In that particular form of Maxwell's Electro- 

 Magnetic Theory of Light in which the media are supposed 

 to be non-magnetic, we have exactly the same analytical con- 

 ditions, so far as plane waves in a non-absorbent medium are 

 concerned, as in three other theories of light, viz.: (1) Fresnel's 

 (2) MacCullagh's, (3) Lord Kelvin's (contractile aether). To 

 be more precise : — (1) The electromagnetic displacement, D, 

 is the exact mathematical equivalent of Fresnel's displacement; 

 (2) the electromagnetic M.M.F., H (or, what is the same in 

 the present case, the magnetic induction B), is the exact 

 mathematical equivalent of MacCullagh's displacement; (3) 

 the electromagnetic E.M.F., E, is the exact mathematical 

 equivalent of Lord Kelvin's displacement. 



In the second and third cases the mathematical equivalence 

 extends, not only to the properties of plane ivaves within a medium, 

 but also to the boundary conditions between two media. ± his is 

 true in the first case also, in so far that the laws of reflexion and 

 refraction at the boundary of isotropic transparent media, 

 which result from the two theories, are the same. 



If only for this extraordinary property of being mathe- 

 matically equivalent to so many distinct and celebrated 

 theories, the electromagnetic theory is worthy of study. 



It is to be remarked that it is only a particular form 

 (certainly a very general particular form) of the electro- 

 magnetic theory which has these wide and exact mathe- 

 matical analogies with other theories. Mr. Heaviside has 

 taught us to look, in Electromagnetism, for a symmetrical 

 dual interpretation of results, the electric and magnetic 

 quantities providing the duality. It may be anticipated that 

 in the general electromagnetic theory of light, insight will be 

 obtained by adhering to the symmetrical dual interpretation. 



This first part of the present paper is mainly concerned with 

 geometrical properties of the wave-surface exhibiting this 

 symmetry. Incidentally a new presentation of the funda- 

 mental analysis will be given. 



Four geometrical theorems are enunciated, and immediately 

 after the enunciation of each, some of its obvious geometrical 

 consequences are detailed. The proofs of the propositions are 

 then given. 



