ON THE ACTION OF MAGNETISM ON LIGHT. 371 



In contrast with these explanations, the real reason why the theories 

 of MacCullagh and KirchhofF agree in their results will now be stated. 

 It is simply that, when £, jj, 'C are functions of a linear function of x, y, z 

 and t, and therefore are the displacements in a plane-wave of some form, 

 the unmodified expression for Kirchhoff's energy-function P reduces to 

 MacCullagh's energy-function U, the various Jacobian expressions 

 d(ri, ii)/d(y, z), &c., contained in it being all null. For a wave with a 

 spherical or other curved form of front, these terms would not thus dis- 

 appear ; and the boundary conditions could not, I think, be reduced to 

 the proper number by Kirchhoff's process. The conclusion to be drawn 

 from this would be as before mentioned, not that reflexion cannot be 

 explained, but that Green's expression for the energy, as employed by 

 KirchhofF, is untenable. 



We have seen that a labile aether gives results conjugate to, but 

 not the same as, those of the rotational sether corresponding to Mac- 

 Cullagh's equations. It is also known that Neumann's simple theory 

 which can be expressed by means of rays, without technical considera- 

 tions of elasticity, leads to the same results as MacCullagh's ; and we 

 now see that KirchhoflP's method would lead to the same result. Now 

 the elastic solid theory of Kirchhofi" is in its elements just the same as 

 the labile jsther elastic solid theory ; and yet Kirch hoff' gets a different 

 result out of it. This demonstrates still further the faultiness of his 

 procedure : he is not entitled to throw away the Jacobian terms in the 

 energy because they happen to be null for the plane- wave kind of motion 

 which he assumes to be the only one to which the reflexion will give 

 rise ; though he happens to be led to the correct result by equilibrating 

 them, as he can clearly do for this particular case, by extraneous surface 

 tractions of null activity. Further, it thus appears that, according to the 

 form he takes for his extraneous forces, he can arrive from the same 

 data at either of two conjugate theories of reflexion. 



Mechanical Illustrations of MacCullagh's Theory. 



30. The conclusions here arrived at naturally tempt one to pursue 

 the invention of mechanical illustrations of the aether. Lord Kelvin 

 proposes to realise and illustrate his labile contractile tether by a homo- 

 geneous mass of foam free from air. Such a medium, when distorted, will 

 have its equilibrium disturbed, and will tend to recover itself ; when 

 uniformly compressed it will exhibit volume-elasticity. But when it is 

 compressed in one direction only in plane layers, there will be no 

 tendency to recover : its Young's modulus will he null, and so there will 

 exist a fixed ratio between its compressibility and its rigidity, an inter- 

 esting result which it would be rather difficult to investigate directly. 

 Longitudinal waves will thus not be propagated in the medium. 



We have also two types of Lord Kelvin's gyrostatic aethers, one of 

 them with pure rotational elasticity and no compressional or distortional 

 elasticity, the other incompressible but with no distortional elasticity ; 

 either of them will represent MacCullagh's equations. A mechanical 

 realisation of an aether of the second kind has been proposed by Fitz- 

 Gerald as consisting of a web of long vortex filaments, interlaced together 

 in homogeneous frictionless incompressible liquid, with any desired iso- 

 tropic or crystalline quality : but even if we could be assured that such 

 a system could subsist, and not be at once hopelessly entangled and 



B B 'i 



