K3) 



Royal Irish Academy, 231 



Now if we take the variations of these expressions, and substitute 

 them in the value of 2 v derived from equation (2), then multiply by 

 dv' dy' dz', and integrate between the limits s' = and 2;' = go, 

 neglecting to take account of the latter limit, as well as of the in- 

 tegrations with respect to ocf and y', of which both the limits are 

 infinite, we shall get, in the equation which holds at the separating 

 surface, a term of the form 



/fdx'dy'iddt,' -Fdr,'),' (4) 



where 



v — a"^ I -^' --^ ) cos a + 62 / ^ --! I cos a' 



\dz dy ) \dx dz/ 



, , /dl drji\ ,, 



+ c~ {- :r ) cos a", 



\ay ax) 



This term, along with a similar but simpler one arising from the 

 ordinary medium, must be equal to zero ; and as the variations S ^' 

 and t] are independent, this condition is equivalent to two. More- 

 over, the quantities |' and tj' are to be put equal to the correspond- 

 ing 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, 'il , jj', 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 ^' likewise retains its value. These two consequences 

 were used as hypotheses by the author in his former paper, and ac- 

 cordingly all the conclusions which he has drawn in that paper will 

 follow from the present theory also. 



>(5) 



