377 
When there are two contiguous media, and the light 
passes out of one into the other, suppose out of an ordinary 
into an extraordinary one, and we wish to determine the 
laws of the reflected and refracted vibrations, it is only 
necessary to attend to the double integrals in the equation 
of limits ; but the integrations must now be performed 
with respect to other coordinates. Taking the separating 
surface of the two media for the new plane of xy, the axis 
of « being in the plane of incidence, let the principal axis 
x of the crystal make with these new axes the angles 
a, 8, y, while the principal axes y and z, in like manner, 
make with them the angles a’, 3’, 7, and a’, 3”, y”, res- 
pectively. Then, marking with accents the quantities 
relative to the new coordinates, we have 
dn_ a (- a, )e os'a + (<- +.) cos [3 
dz dy~ \dz” dy | 
(3 ide a) bei | 
ap aera one) eo 
dé’ dn! 
+(e da /) £08 
$£-G S)mes(Es$)mr 
+ (Gyo Ge) er 
Now if we take the variations of these expressions, and 
substitute them in the value of dy derived from equation 
(2), then multiply by da‘dy‘dz’, and integrate between the 
limits z= 0 and s’= @, neglecting to take account of the 
latter limit, as well as of the integrations with respect to 
x 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 
