158 Prof. D. N. Mallik 



on 



Thus the expressions to be integrated in the equations 

 expressing Hamilton's principle and Fermat's law are quite 

 distinct, and cannot be treated as identical. Dr. Mallik's 

 conclusion appears to me therefore not proved. 



4. That this must be the case may be seen in the following 

 manner. In the various elastic solid theories it has been 

 shown that certain assumptions as to the elastic constants 

 will lead to results consistent with the laws of refraction r 

 and so of propagation through an isotropic medium in which 

 the refractive index varies from point to point. This is 

 sufficient to ensure that Fermat's law shall be completely 

 obeyed ; and the analytical condition that Fermat's law 

 shall be obeyed will therefore give no additional result 

 regarding the potential energy. 



In the corresponding case of sound also the velocity 

 of the particles of air and of the velocity of sound are 

 obviously quite distinct. So that though Fermat's law of 

 least time also holds as a consequence of the ordinary sound 

 equations * there is no fresh information to be deduced from 

 Fermat's law. 



5. All this argument applies merely to dynamical theories 

 of light, to which alone is there a satisfactory justification 

 for applying Hamilton's principle. It cannot therefore be 

 convincing to those who regard tbe electromagnetic theory, 

 with certain modifications, as holding the field. It is quite 

 true, as Professor Mallik says in section 14 of his paper, 

 that we can {i interpret the intimate nature of the electric 

 field" as possessing the same qualities as MacCullagh's 

 medium ; but this is different from being " led to conclude 

 that all energy is kinetic." 



Gilbert T. Walker. 



XIV. On Fermat's Law. 



Presidency College, Calcutta, 

 27th April, 1918. 



To the Editors of the Philosophical Magazine. 

 Gentlemen, — 

 rj THROUGH the courtesy of Dr. Gilbert Walker, I have 

 X been permitted to see in advance the letter which he 

 has written to the Philosophical Magazine criticising a 



* See Rayleigh's ' Sound, 'vol. ii. §289 in the Second Edition. 



