444 Mr. J. Larmor. A Dynamical Theory of [Dec. 7, 



down the unknown and in several ways mysterious and paradoxical 

 properties of the luminiferous medium to be the same as those of an 

 ordinary elastic solid. 



The form of MacCullagh's energy-function was derived by him 

 very easily from the consideration of the fact that it is required of it 

 that it shall produce, in crystalline media, plane-polarised waves pro- 

 pagated by displacements in the plane of the wave front. Though 

 he seems to put his reasoning as demonstrative on this point, it has 

 been pointed out by Sir George Stokes, and is indeed obvious at 

 once from Green's results, that other forms of the energy-function 

 besides MacCullagh's would satisfy this condition. But the important 

 point as regards MacCullagh's function is that it makes the energy 

 in the medium depend solely on the absolute rotational displacements 

 of its elements from their equilibrium orientations, not at all on its 

 distortion or compression, which are the quantities on which the 

 elasticity of a solid would depend according to Green. 



Starting from this conception of rotational elasticity, it can be 

 shown that, if we neglect for the moment optical dispersion, every 

 crystalline optical medium has three principal elastic axes, and its 

 wave- surf ace is precisely that of Fresnel, while the laws of reflexion 

 and refraction agree precisely with experiment. Further, ib follows 

 from the observed fact of transparency in combination with dis- 

 persion, that the dispersion of a wave of permanent type is properly 

 accounted for by the addition to the equations, therefore to the 

 energy-function, of subsidiary terms involving spacial differentiations 

 of higher order. To preserve the medium hydro dynamically a perfect 

 fluid, these terms also must satisfy the condition that the elasticity of 

 the medium is thoroughly independent of compression and distortion 

 of its elements, and wholly dependent on absolute rotation. It can 

 be shown, I believe, that this restriction limits the terms to two 

 kinds, one of which retains Fresnel's wave surface unaltered, while 

 the other modifies it in a definite manner stated without proof by 

 MacCullagh [; but the first terms depend on an interaction between 

 the dispersive property and the wave motion itself, while the second 

 terms involve the square of the dispersive quality. It seems clear 

 that the second type involves only phenomena of a higher order of 

 small quantities than we are here considering December 7, 1893] ; 

 thus an account of dispersion remains which retains Fresnel's wave 

 surface unaltered for each homogeneous constituent of the light, 

 while it includes the dispersion of the' axes of optical symmetry in 

 crystals as regards both their magnitudes and directions results 

 quite unapproached by any other theory ever entertained. 



In this analysis of dispersions, all terms have been omitted which 

 possess a unilateral character, such as would be indicated in actuality 

 by rotatory polarisation and other such phenomena. The laws of 



