116 On the Reflexion and Refraction of Light. 



of equality of normal displacement and of surface traction 

 parallel to Oz cannot be satisfied without some surface action. 

 The elimination of the terms expressing this surface-action 

 from these two equations of condition gives us our equation 

 (16). On the Electric theory, if we suppose a surface dis- 

 tribution of variable density possible, terms would come into 

 the two surface conditions already mentioned, depending 

 on this distribution ; we should thus have two equations 

 corresponding to our (4) and (7), and the elimination of 

 the surface-density from these would give us an equation 

 equivalent to (16). 



It is perhaps worth while to remark that equation (9) or 

 (16) holds, even though the constant A be not zero. For 



since u is continuous across the surface, so is also j- ; and 



dv du . ,. ., . dv du . & 

 since -= — |- -y- is continuous, we see that - —is also con- 



y dv du y 



tinuous. But we have -r^= -f^; and hence, in the expression 



dv du 

 for the continuity of -^ — — , the terms involving the pres- 



sural wave will not occur, and this condition will give us 

 equation (9) at once. But if A is not zero, (5) will be 

 modified, and becomes 



Scos/3+S 1 cosA+^^=S / cos/5' + S^cos/S ,, + 5^^; (23) 



A,t A l 



while the continuity of Nj leads, if we assume A to be the 

 same in both media, to 



A(* f + ^)-2B^=A(^' + ^-')-2B^' ; . (24 ) 

 \dx dy/ dy \dx dy ' dy v J 



which since v, and therefore -y-, is continuous, reduces to 



du du' 



Tx = lM ( 25 ) 



We also require the equation of motion for the pressural wave 

 and the problem is much more complicated ; it has been 

 solved for two isotropic media in Sir W. Thomson's paper, 

 and the solution in the present case must proceed along the 

 same lines. 



