Equations of Electrodynamics for Moving Media. 43 



But also 



tpdq — E.dt = dS, (5) 



where S is the principal function [S(<y, t, a)} for the original 

 variables. Adding (3) and (5) we get 



2p'dg'-(H + U)<fc=<*(W + S). . . . (G) 



But since -BW/d^=BS/B?, we. have d(W-r S)/dg=0, that 

 is # does not appear in the function W + S, and so we get 

 the canonical equations 



dp[_ _d(H + U) ^'_ d(R + U) 



The principal function is now W + S. 



IV. The Equations of Electrodynamics for Moving Ponderable 

 Media and the Principle of Relativity. By H. R. Hasse, 

 M.A., M.Sc, Fellow of St. John's College, Cambridge ; 

 Fielden Lecturer in Mathematics, Manchester University 



* 



I. 



rf^HE Theory of Relativity, as applied to Maxwell's equa- 

 JL tions of the aether, rests on the work of Larmor and 

 Lorentz, who independently proved that the equations were 

 invariant under a certain space-time transformation. The 

 first attempt to extend this result to the case of ponderable 

 media was made by Lorentz f, who in 1904 applied the 

 same transformation to the case of a non-magnetizable 

 dielectric in an investigation as to the possibility of there 

 being a positive result for the Michelson-Morley experiment 

 in the case of propagation through a dielectric. 



The discovery of a complete scheme of electrodynamical 

 equations which shall satisfy the Principle of Relativity for 

 the case of uniform translation is due to Minkowski $, who 

 obtained a scheme of equations of the electromagnetic field 

 which differed from that given by Larmor and Lorentz. 

 These latter equations were, however, soon afterwards shown 

 to be also consistent with the demands of this principle. 



In this part of the paper we shall examine the way in 

 which Minkowski obtained his scheme ot equations, in order 

 to find a satisfactory method of making use of the principle 

 of relativity. 



* Communicated by the Author. 



t Proc. Acad, of Sciences of Amsterdam, 1904, p. 809. 

 % Gottinger Nachrichten, 1908, reprinted in Math. Ann. vol. lxiii. 

 (1910). References will be made to the reprint. 



