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 x being in the plane of incidence, let the principal axis 

 x of the crystal make with these new axes the angles 

 a > fi, y, while the principal axes y and z, in like manner, 

 make with them the angles a, B', y, and a", B", y", res- 

 pectively. Then, marking with accents the quantities 

 relative to the new coordinates, we have 



d n dl fd n ' d?\ . fdt dl\ ., 



dy \dz dy 

 '0 

 t3y 



, (dV dtf\ 



d A _ ^ - ( d JL - d Jl\ cos a' + (S - d %-\ cos B' 

 dx ds \dz' dy'J \dx dz'J 



/^'_^]'\ cos ', 

 \dy dx' ) 



(3) 



<dy 

 dZ_d 1 _W_ dV\ cog ' fdV f dj\ cog 

 dy dx ~ \dss' dy'J \dx dz'J 



. fdt dri'\ „ 



+ \dy-'-d^') C0 ^' 



Now if we take the variations of these expressions, and 

 substitute them in the value of Sv derived from equation 

 (2), then multiply by dx'dy'dz, and integrate between the 

 limits ss'zz.0 and % ■=. oo, 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 



