Crystalline Reflexion and Refraction. 169 



in each medium, we should get the laws of propagation in each. 

 But we are not now considering these laws, and we need only 

 attend to the double integrals produced by the operation afore- 

 said. 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 Q y^\ and this part 

 must be separately equal to zero, since the conditions to be ful- 

 filled at the plane of X Q y Q are independent of anything that 

 might take place at other limiting surfaces, if such were sup- 

 posed to exist. Collecting therefore the terms produced by in- 

 tegrating with respect to s , and observing that a negative sign 

 must be interposed between those which belong to different 

 media, we get 



y Q (F' S' - *' &i'o) - If dx, dy Q (Q$?' - P^o) = 0, (18) 

 where 



P = a 2 lX + WmY + c z nZ, Q = a*?X + b*m'Y + c*n'Z. (19) 



In each of these equations it is understood that s = 0. But 

 when z = 0, we have obviously 



r, - r, * - ," 



and therefore 



Sc, Q = o? o) 017 = Sj o ; 



so that the equation (18) becomes 



which, as the variations rf' and dif Q are arbitrary and indepen- 

 dent, is equivalent to the two equations 



X\ = P, Y\ = Q. (21) 



Thus, to find the relations which subsist among the vibra- 

 tions' incident, reflected, and refracted, at the common surface 

 of two media, we have four conditions, expressed by the equa- 

 tions (20) and (21) ; and these conditions are sufficient to deter- 

 mine the reflected and refracted vibrations, when the incident 



