378 



ss^wc^r-w), (4) 



where 



cly 



P zz ar ( - — I cosa 4- or \ .-z r- cos a 



\az dyj \dx dssJ 



\dy dx) 



2 fdr] dZ\ _ 2 /o?£ d%\ , 



Q = a {d-,-d7j) C0S ^ + b Kdx^Tj C0 ^ 



dy. 



(5) 



<dy 



This term, along with a similar but simpler one arising 

 from the ordinary medium, must be equal to zero ; and as 

 the variations ££' and St/ are independent, this condition 

 is equivalent to two. Moreover, the quantities X and ij' 

 are to be put equal to the corresponding quantities in the 

 other medium, and thus we have two more conditions, 

 which are all that are necessary for the solution of the 

 problem. 



The four conditions may be stated by saying, that each 

 of the quantities p, q, £', t\ retains its value in passing out 

 of one medium into another. Hence it is easy to show 

 that the vis viva is preserved, and that Z,' likewise retains 

 its value. These two consequences were used as hypotheses 

 by the author in his former paper, and accordingly all the 

 conclusions which he has drawn in that paper will follow 

 from the present theory also. 



It will be perceived that this theory employs the general 

 processes of analytical mechanics, as delivered by Lagrange. 

 The first attempt to treat the subject of reflexion and re- 

 fraction in this manner was made by Mr. Green, in a very 

 remarkable paper, printed in the Cambridge Transactions, 

 vol. vii. part 1. After stating the dynamical principle 

 expressed by equation (1), (though with a different hypo- 

 thesis respecting the density of the ether,) Mr. Green ob- 



