366 EEPORT— 1893. 



forces act except sucli as are called forth by their relative displacements ; 

 while MacCnllagh takes for the potential of the forces in operation on 

 the elements of the sether an expression which does not agree with the 

 potential of the forces called into play by the relative displacements of 

 the parts of an elastic solid. Thus in the theory of MacCullagh, if we 

 are to treat the tether as an elastic solid, we must treat it as one which 

 is acted on by forces in addition to those called into play by its elasticity. 

 Yet of these other forces it may be pi^oved from the energy-function 

 adopted by MacCullagh that, taken throughoat a portion of the aether in 

 a homogeneous body, they reduce to tractions which operate on its surface. 

 "We can therefore assert, that the theory of MacCullagh rests on the 

 hypothesis that on the elements of the asther no forces act except such 

 as are derived from its elasticity ; but on the surfaces which form the 

 boundaries of heterogeneous media tractions are imposed which have 

 some other origin. And such tractions must also be contemplated by 

 Neumann's theory ; their function is that we are by their aid empowered 

 to leave the compi'essional wave out of consideration, just as happens in 

 the former theory : they must exist, in order that compressional waves 

 may not be set up in the reflexion and refraction of light- waves. The 

 two theories compai-ed can thus be seen to be in complete accord. I 

 propose to myself to lay before the Academy a treatment of the question 

 from the standpoint of these theories which, I think, is more general and 

 more comprehensive than those that have been given hitherto.' These 

 imposed interfacial forces are restricted merely to satisfy the condition 

 that they shall do no work on any element during the actual displace- 

 ments of the media ; they are considered by KirchhofF to be ' tractions 

 from without (fremJen Druclchrufte) which act on an element of the 

 interface, tractions which, we are accustomed to assert, arise from the 

 forces which the ponderable parts of the two media sustain from 

 the aether.' ' The forces contemplated by KirchhofiP's principle, in 

 order to allow of the condition of incompi-essibility being satisfied, are 

 thus only interfacial tractions, which form an equilibrating system in so 

 far as they do no work in any displacement actually contemplated. 

 According to the elucidation and extension of MacCullagh's pi'inciple 

 which is here proposed, they should be taken to be a system of pressures 

 distributed throughout the media, which do no work for the displace- 

 ments actually contemplated, and which are in so far equilibrating. 

 These pressures will be discontinuous at an interface ; and will hence 

 modify the boundary conditions in the same manner as Kirchhoff's 

 extraneous forces. 



26. In the account of Kirchhoff's principle given by Volkmann,'^ the 

 view is propounded that such a principle is necessary because the equa- 

 tions of an elastic solid medium, with the addition of a pressure intro- 

 duced in the manner indicated above (§ 11), will not lead to an account 

 of reflexion which is in accordance with experiment. Quoting from 

 Kirchhoff's lectures on Optics (p. 143), ' We have to recognise that the 

 elasticity of the ajther is different in the various transparent media, 

 different in glass, for example, from what it is in empty space. We are 

 not in a position to form for ourselves a clear representation as to how 

 the alteration of the elasticity of the eether in glass is brought about ; but 



' G. Kirchboff, loc. cit. ; Gesainmelte Ahhandl., p. 367. 

 2 P. Volkmann, Theorie des Zichtcs, 1891 :§ 76. 



