Theory of a Contractile ^Etliev to Optical Problems. 535 

 Also we find 



where 



R2 _ ^V 9 1 



and 9 ^ ( 58 ) 



(W«-V 2 > 2 J 



Now we notice that & is a quantity which depends on 7 2 and 

 is small therefore when y 2 is small. In a transparent body 

 k must be practically zero, and hence we infer that for a 

 transparent body 7 2 is extremely small. The only reason for 

 retaining the term at all lies in the fact that if we put 7 2 = 0, 

 then for the value » 2 =y 2 we have the ratio XJ /u infinite. 



Taking, then, a case in which 7 2 is zero, and remembering 

 that our solution fails for the critical value v 2 of n 2 , we have 

 in general U /w small, and 



Y 2 ~B Bp 2 {n 2 -v 2 ) ^ DJ) 



Now let Y be the velocity in vacuo of light of the frequency 

 n ; X its wave-length. 

 Then 2ttV 



n= -XT- 

 Pot 



2ttV 



V= TCT' 



and substitute in (59) 



V 2 B + Bp 2 \ 2 -\ 1 2 ^ V) 



Also, if fi = V / V = the refractive index. Since V 2 = B/p , 



p^L + JL V_ (61) 



Po P0P2 X 2 -X, 2 K y 



The quantity p/p is the square of the refractive index for 

 waves of infinite length ; put it fi* 3 and write C for p ,2 /p p 2 . 

 Then 



*-*?+sJ=k?-- •••••• (62) 



This is Ketteler's dispersion formula, which he has proved 

 agrees well with the results of experiment over a long range 

 of values of X*. 



By supposing, as is done by Sir W. Thomson in the Balti- 

 * Ketteler, Wied. Ann. xii. pp. .303, 481, xv. p. 336, and elsewhere. 



