ON THE REFLEXION AND REFKACTION OF LIGHT. 255 



In like manner we get 



+ the triple integral ; 



since it is the least value of x which belongs to the surface of 

 junction in the lower medium, and therefore the double integrals 

 belonging to the limiting surface must have their signs changed. 



If, now, we substitute the preceding expression in (3), equate 

 separately to zero the coefficients of the independent variation 

 $u, &v, Sw, under the triple sign of integration, there results for 

 the upper medium 



d*u A d (du , dv , dw\ n (d z u d z u d fdv dw\) 



P^r=A -j--\-j- +-J- + y')+-5lj-i +-7-5 JT- (-T- + -T- r; 

 r dtf dx \dx dy dz] \dy* dz* dx \dy dzj)' 



d 2 v . d fdu dv dw\ (d*v d 2 v d (du dw 



P =A-J-. T- + ^- + ^- +- 5 ij-i + TT- j-*\-j- + ~i- 



r dt 2 dy \dx dy dz) (dx* dz 2 dy \dx dz 



^ df dz\dx dy dz) (dx 2 dy 2 dz'\dx dy)) 1 



and by equating the coefficients of 8^, Sv,, $w,, we get three 

 similar equations for the lower medium. 



To the six general equations just obtained, we must add the 

 conditions due to the surface of junction of the two media ; and 

 at this surface we have first, 



u = u^ v=v t , w = w j} (when x = 0) ,.(5); 



and consequently, 



