ON THE ACTION Oi^ MAGNETISM ON LIGHT. 365 



light may be thrown by the consideration of a quite similar difficulty 

 in the dynamics of actual elastic systems, which has recently occupied 

 the attention of several mathematicians. The vibrations of a curved 

 elastic plate, in fact of a bell supposed of small thickness, have been 

 worked out by Lord Rayleigh,' simply from the energy -function of the 

 plate. The plate being thin, it can easily be deformed by bending ; on 

 the other hand to stretch it sensibly would be very difficult. For this 

 reason the energy- function is formed by Lord Rayleigh on the assumption 

 that the plate is perfectly inextensible, so that terms depending on exten- 

 sion do not occur in its expression. Some years subsequently it was 

 pointed out by Love "^ that this treatment does not allow of all the elastic 

 conditions at the boundary of the plate being satisfied. Now on the 

 principles here expounded the adjustment of these terminal conditions 

 would be made by tensions in the plate, which, owing to the very rapid 

 velocity of propagation of extensional disturbances, practically obey at 

 each instant an equilibrium theoiy of their own, and at the same time 

 involve the play of only a negligible amount of energy owing to the 

 magnitude of their elastic modulus. If the plate were quite inextensible 

 these tensions would be absolutely in equilibrium at each instant, and 

 the energy-changes involved in them would be null. And this view is, 

 1 believe, in agreement with the mode of explanation now generally 

 accepted for that problem.^ The solution of the problem of vibration of 

 a bell may thus be derived, as regards all things essential, from the 

 energy-function of the bending alone, combined explicitly or implicitly 

 with the geometrical condition of absence of extension. 



Critique of Kirchhoff's Theory, 



25. The principle implied in MacCullagh's analysis is claimed to be 

 identical, in its results if not in theory, with a hypothesis adopted by 

 Kirchhofi" in his discnssion of crystalline reflexion,'* which is commonly 

 quoted by German authors nnder the title of Kirchhoff's principle. Its 

 author employs it avowedly as a formal mathematical representation of 

 assumptions made explicitly by F. Neumann, and tacitly he says by 

 MacCullagh, in their theories, which it is the object of his memoir to 

 reproduce and amplify. He attempts no dynamical justification of its 

 use ; on the other hand he rather formulates it as an additional hypo- 

 thesis. At any rate it has been treated as a hypothesis by Kirchhoff's 

 followers in Germany, while its validity is suspected by some other 

 writers who have considered the subject. The explanation of Kirchhoff 

 himself in the introductory paragraph of his memoir, in comparing 

 Neumann's and MacCullagh's theories, is here reproduced in a free 

 translation. * Yet at the first glance the points of departure of the two 

 theories would appear to be ditferent, even diametrically opposed to each 

 other. For Neumann starts from the view that the a3ther in respect of 

 light-vibi'ations comports itself as an elastic solid, on whose elements no 



' Lord Rayleigh, ' On the Infinitesimal Bending of Surfaces of Revolution,' Proc. 

 Land. Math. Soc, xiii. 1882. 



2 A. E. H. Love, Phil. Trans., 1888. 



= Cf. A. E. H. Love, Treatise on Elasticity, vol. ii. 1893, § 349. 



' G. Kirchhoff, ' Ueber die Reflexion und Brechung des Lichts an der Grenze 

 krystallinischer Mittel,' Ahh. dvr Berl. Aliad., 1876 ; Gesavimclte Ahhandl., p. 352. 



