231 
Royal Irish Academy, 
d tj 
dz 
d_K 
dx 
dy 
Now if we take the variations of these expressions, and substitute 
them in the value of ^v derived from equation (2), then multiply by 
dx^ dy^ dz\ and integrate between the limits z^ = 0 and z’ = co, 
neglecting to take account of the latter limit, as well as of the in- 
tegrations with respect to xf 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 
ffdxUy^ (qiH' - 
where 
(4) 
;>(5) 
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 ^ jj' 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, tj', 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. 
