AND REFRACTION OF LIGHT. 17 



In like manner, if in the second medium we make 



' dx dy ' 



(15). 



v - d( f > ' d ^> 

 ' dy dx ' 



we get to determine <p, and >|/ / the equations 



df *' \da* dy 2 

 ^ _ * * ( d l±i x £*< 



(16). 



df " \dx* dtf 



and as we suppose the constants A and B the same for both media, 

 we , have 



7, g,' 



For the complete determination of the motion in question, it will be 

 necessary to satisfy all the conditions due to the surface of junction of 

 the two media. But, since w = and w t = 0, also, since u, v, u t , v, are 

 independent of as, the equations (5) and (6) become 



u = u t , v = »,; 



\dx dy) dy \dx dy I dy 



du dv _ du, dv, 

 dy dx dy dx ' 



provided x = 0. But since x = in the last equations, we may differ- 

 entiate them with regard to any of the independent variables except x, 

 and thus the two latter, in consequence of the two former, will become 



du _ du, dv _ dv, 



dx dx' dx dx' 



Substituting now for u, v, &c, their values (13) and (15), in <p and \f/, 

 the four resulting conditions relative to the surface of junction of the 

 two media may be written, 



Vol. VII. Part I. C 



