Format's Law. 159 



certain interpretation of Fermat's law which I have given. 

 He has very kindly suggested to me that I might offer 

 a reply, if I wished, in the same number of the Magazine. 

 I trust, therefore, that you will kindly permit me to offer 

 not a reply, but some explanations on the subject. 



1. It will be useful in the first place to state my exact 

 point of view. 



We have the dynamical equation 6^(T — Y)dt = which 

 gives a complete account of the motion of a dynamical 

 system — in this case, the disturbed optical medium. 



We have also the equation S\dt = (the dynamical sig- 

 nificance of which requires investigation). 



Are these independent of each other? If so, the second 

 equation can only be the equation of constraint. No such 

 constraint can, so far as we can see at present, be well 

 associated with the medium considered. 



If no such constraint can be postulated, we can only 

 regard the second equation as identical with the first, in 

 this particular case. 



2. Now in order that we should be justified in doing so,, 

 it is necessary to admit that t has the same meaning in both 

 Hamilton's principle and in Fermat's law. 



3. Dr. Walker maintains that this is not permissible as " in 



Format's law we have 8 \dt — or Si— =0, where the 



co-ordinates are no longer those of particles of matter as in 

 Hamilton's principle, but successive points on the ray as the 

 light-wave travels along, and the velocity of the ray is 

 entirely different from the velocity of the individual material 

 particles whose motion constitutes the light." In other 

 words, Dr. Walker bases his argument on (2), which is 

 derived from (1) by a mere analytical transformation which 

 apparently imposes limitations (from the point of view of 

 the present line of argument) on equation ( 1), not necessarily 

 involved in it. In fact, the second equation may well be 



taken as 8 I -^ where 0'= — and may, as such, be held 



to give information regarding d) and </>', whatever tJiese may 

 he (not merely s and v), so long as these quantities are 

 related in any manner to the phenomenon of light propaga- 

 tion. But the particular co-ordinates involved in <p and <£' 

 or their nature cannot well be regarded as alone implied in 

 (1). I conceive, therefore, that the present line of argument 

 is not crucial against my theory. 



