Light from Transparent Matter. 95 



(10) is replaced by 



2a'bcf> + {b*-a*)X=2a;bcl> l +(b*-a?)f l , 

 or, by (21), 



X= S#*< = ^ ( 23 ) 



From (21), (22), 



Y_a, &y«-l) *«-*,« 



^"a a a\k 2 -dk 2 l ; 



From (24) we may fall back on Green's corresponding equa- 

 tion (13) by putting £ = 0, # y = 0, kji k=/j, : 1 -, but Cauchy 

 supposes, on the contrary, that k 2 , kf are very large in compa- 

 rison with b 2 } and writes 



a' = ik, a/^—ik, 

 which convert (24) into 



yjr t a a 



\k k t J 



Cauchy further takes 



2tt/1_ J_\ 

 \\k kj~ 



so that, since — sin 6 = b } the solution of the problem is 



A 



t-{;-k^-i)J.-»}*.) • • (25) 



It may, however, be remarked that Cauchy has no right to 

 suppose that e is a constant for the rays of different wave-lengths. 

 In fact if k and k, are constants, e varies inversely as X; so that 

 the same objection arises here as in the theory of Lorenz. The 

 only difference between (25) and (12), (13) lies in the substitu- 

 tion of — esin# for M. It is therefore unnecessary to write 

 down the results corresponding to (15), (16), (17), (18). 



But what I wish particularly to point out is the extraordinary 

 differential equation satisfied by <p. By differentiating the ex- 

 pression for <f> and substitution in (20), we find 

 d 2 <j> d 2 ^ _ _ *» d 2 <j> 

 dx 2 dy 2 c 2 dt 2 



I am at a loss to understand how any mechanical theory of im- 

 perfect elasticity could lead to such an equation. If we were to 

 speculate as to the most probable form of the equation of motion, 

 we should perhaps give the preference to 



di* T di~ 9 \dn* dy*)' 



