AND REFRACTION OF LIGHT. 11 



d*u . d_ idu dv dw\ _ (d 2 u d"u d idv dw\ 1 

 p ~df ~ A dx'\dx + dy + dz) + [dy 1 + ~d% 2 ~ dx' Xjdy^ ~d%)y 



d 2 v . d idu dv dw\ -n\d 2 v d 2 v d idu dw\) 

 9 ~dt 2 ~ dl/'\dx' + dy + ~dj) + \dx 1 + dz T ~d^'(d^ + 'd^)f' (4); 



d 2 w _ . d idu dv dw \ „ ( d 2 w d 2 w d fdu dv\i 

 p dt 2 ~ dz'\dx dy d%) \dx 2 dy 2 dz'\dx dy)]'' 



and by equating the coefficients of Su /} Sv t , $w t , we get three similar 

 equations for the lower medium. 



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

 ditions due to the surface of junction of the two media; and at this 

 surface we have first, 



u = u t , v = v t , w = w t , (when x = 0), (5); 



and consequently, 



%u = $u t ; 8v = Sv t ; §w = Sw,. 



But the part of the equation (3) belonging to this surface, and which 

 yet remains to be satisfied, is 



♦/r**{*.(£*&)***(&*&W« 



and as 8u = Su t , &c, we obtain, as before, 



. idu dv dw\ „ idv dw\ . ldu t dv^ dw \ __ „ idv^ dw \ 



\dx dy dz J \dy dz J ' \dx dy dz) ' \dy d% I 



„ idu dw\ _ ldu / dw \ _ 



\d% dx) ~ ' \d% dx J ' 



and these belong to the particular value x = 0. 



B2 



