3-60 On Fer mat's Laic. 



•4. Moreover, it stands to reason that Hamilton's principle, 

 although it directly deals with a certain volume distribu- 

 tion of energy, will naturally lead to equations giving pro- 

 pagation of energy or disturbance, if T and V are appro- 

 priate to such a propagation. The latter equations mar, 

 in this event, involve a set of co-ordinates distinct from 

 that involved in Hamilton's principle and yet we may 

 not argue that these equations are independent of that 

 principle. 



5. It is now necessary to consider another difficulty, which 

 Dr. Walker raises and which has always appeared to me 

 of great importance. We know that Hamilton's principle 

 postulates that the initial and final configuration of a 

 dynamical system are prescribed and the time of transit of 

 the system from the initial to the final configuration must 

 remain unchanged. Now the first condition applies also to 

 Format's law and the second condition may be imposed on 

 it, if we take the time of transit to be that from one wave-front 

 to the next. This, however, may deprive my conclusions of 

 a part of their generality, but only in a manner which it is 

 not possible to decipher at present. 



6. As to the contention that " in Hamilton's principle the 

 co-ordinates are, those of particles of matter," I am doubtful 

 whether this limitation will be universally acceptable. 1 

 rather think, given the forms of T and V, in a medium which 

 is the seat of energy, Hamilton's principle will be applicable, 

 though we may be unable to determine the intimate nature 

 of the constitution of the medium which determines T 

 and V. In order to arrive at these forms, various hypotheses 

 have to be framed, and we thus get various forms of Tand V 

 and corresponding optical theories. From this point of 

 view, the electro- magnetic theory with or without modifica- 

 tions may well be regarded as a dynamical theory. This 

 may meet the difficulty to which Dr. Walker refers in 

 para. 5. 



7. I must admit, however, that even if the identity 

 between Fermat's law and Hamilton's principle can be 

 established, I have not been able as yet to prove that 

 T — V= constant is the only solution. Therefore, although 

 I cannot think of any other solution (T— V=/(£) being- 

 inadmissible, on the principle of energy), I have modified 

 .my conclusion and now content myself with saying (in my 

 book) that we "may take T — V = constant." 



8. This seems to be all the more desirable in view of 

 -what I have stated in paras. 1 and 5. I do not therefore 



