THE REFLEXION AND REFRACTION OF LIGHT. 115 



_ d<p d\\, 

 dx dy ' 



_ d<f> d\js 

 dy dx ' 



u t and v, the disturbances in the lower medium being expressed by 

 similar formulae in rf> t and ^. 



The two last equations of (17) give, since 



x = £•= X 



<p' = M 2 $,', y = m 2 ^/ ; 



<p and <p t being accented for a moment to distinguish between the par- 

 ticular values belonging to the plane (yz) and their more general values 



(j> = e hx <f> and <£, - e" 6 ^/. 



The correctness of these values will be evident on referring to the Memoir, 

 formulae (20), (21), and recollecting that 



b = a' = «;. 

 Hence the first equation gives, since x = 0, 



• • *' " " bWTT) ~dy~ ' ana + btf + 1) dy ' 



Also the second equation may be written, 



dj, _d±,_d$__d(ti . 0* 2 - 1)' dfy, 

 dx dx " dy dy b (m 2 + 1) dy 2 ' 



And since we may differentiate or integrate the equations (17) relative 

 to any variable except x, we get for the conditions requisite to com- 

 plete the determination of ^ and >//,, 



p2 



