378 
§§ da’dy’ (adé’ — Pen’), (4) 
Pa (2-5) cosa + a cos a’ 
Q= a(2-%) cos 3 + 0° (ok) cos [3’ 
+¢ (F- . cos [3". 
This term, along with a similar but simpler one arising 
from the ordinary medium, must be equal to zero ; and as 
the variations 6& and én’ are independent, this condition 
is equivalent to two. Moreover, the quantities & and 1 
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 Pp, @, &, n’ 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 2’ 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- 
