crystalline Reflexion and Refraction. 37 
we have 
x= = lexi’ + Veyi’ a USz' aa 
4o,,4/ Mott 
éy = mbox, + m’ey, + m’oz/,, 
OZ =inbxX, a n bY at ot nex) ‘i 
These expressions for 6x, éy, 62 having been written in formula (3), the 
resulting value of 6V’, as well as the above value of 6V’, is to be substituted in 
the equation (17), and then the right-hand member of that equation is to be in- 
tegrated by parts, in order to get rid of the differential coefficients of the varia- 
tions. When this operation is performed, the triple integrals on one side of the 
equation will be equal to those on the other; and by equating the coefficients 
of the corresponding variations in each medium, we should get the laws of pro- 
pagation in each. But we are not now considering these laws, and we need only 
attend to the double integrals produced by the operation aforesaid. The double 
integrals are together equal to zero; but we are concerned only with that part of 
them which relates to the common limit of the media, the plane of x, y,; and 
this part must be separately equal to zero, since the conditions to be fulfilled at 
the plane of 2, y, are independent of any thing that might take place at other 
limiting surfaces, if such were supposed to exist. Collecting therefore the 
terms produced by integrating with respect to 2, and observing that a negative 
sign must be interposed between those which belong to different media, we get 
Sidr, dy, (x; & — x, ey) — Gda, dy, (Q0&,’ — Pey,’) = 0, (18) 
where 
p=a@lx + b’my +c’nz, gealx + bmy + cn'z. (19) 
In each of these equations it is understood that z,= 0. But when z, = 0, we 
have obviously 
& = 6 No = 0's (20) 
and therefore 
Le = (Ee On == 0N,s 
so that the equation (18) becomes 
Wade, dy, {(¥, — a) 6&, — (x; — P) &y)} = 0, 
