368 KEPORT— 1893. 



medium being incompressible, lie did not take account of sucb waves, and 

 so was in difiBculty with bis boundary equations. He cut the knot by 

 assuming that the displacement is continuous across the interface, in 

 other words that there can be no rupture of material continuity ; and by 

 omitting altogether all conditions of continuity of stress, replacing them 

 by the principle that there is no loss of energy in the act of refraction and 

 reflexion. This, as Kirchhoff remarks, is equivalent to an admission that 

 the equilibrium (or vibrational motion) of an indefinitely thin layer, 

 including in it the interface, is maintained by the aid of forces introduced 

 somehow from outside the vibrating system ; but that, as the energy of 

 the incident light is accounted for exactly by that of the reflected and 

 refracted light, these forces must be subject to the condition that they do 

 no work on any element of this surface layer in the displacements to 

 which the medium is actually subjected during the motion. On this 

 basis Neumann obtains Fresnel's equations of reflexion, by aid of the 

 hypotheses that the displacement of a linear wave is in the plane of 

 polarisation, and that media differ optically in elasticity but not in density. 



As we have seen, Kirchhoff adopts and expounds the method initiated 

 by Neumann for getting over the boundary difficulty. But his main 

 argument is that if we do not assume surface forces from without we are 

 helpless, that such forces exist, as is inferred from molecular theory, but 

 that all we know about them is that in their play they cannot absorb any 

 of the energy of the light. His method of procedure would therefore be 

 to assume the most general possible type of such forces subject to this 

 one condition, and then try by special assumption to adjust them to the 

 final result he desires. There is clearly no dynamical validity in this, it 

 is purely empirical ; the surface forces may really be subject (as we shall 

 see, are subject) to other unknown laws as well, which will not, with the 

 assumed energy-function of the medium, allow of the desired solution. 

 The process would then only prove that the assumed energy^function is 

 untenable. 



28. The correct method is the one indicated above. The energy of 

 the medium is associated with the medium in bulk, is located in its 

 elements of volume. In Gauss' theory of capillarity it is true that inter- 

 facial energy is contemplated, but that is only the actual excess or defect 

 of the energy in the very thin layer of transition over what its amount 

 would be if the transition was supposed sharp and the density of the 

 energy in the elements of each medium near the surface were unaltered 

 by the neighbourhood of the other medium. It is this portion of the 

 energy that produces superficial effects such as surface-tension, though 

 owing to the thinness of the interfacial layer it forms only a very minute 

 fraction of the whole energy, the distribution of the other part being uni- 

 form. Now the propagation of vibrations across the interface is an affair 

 of the redistribution of the enei'gy of the medium en masse ; if we make 

 the ordinary optical hypothesis that the layer of transition is very thin 

 compared with the length of a wave, we may be certain that there is no 

 superficial term of sensible importance in the vibrational energy of the 

 system. The only superficial forces which can come in are, then, those 

 which enter logically in the dynamical analysis of the motion, on the basis 

 of a volume distribution of energy in the medium, the determination of 

 whose form is part of the problem. Until the possibilities of this state- 

 ment of the problem are exhausted, it would appear to be gratuitous and 

 unscientific to assume the existence of unknown surface-forces; and more- 



