﻿Theory of Dispersion. 479 



32. The equations of Helmholtz with the same notation as 

 in §30 are 



mw=a 2 V 2 ,M-f £ 2 (U — u) k 



and ^U=-/3 2 (U-a)-a 2 U- 7 2 U, > 



whence m'co x =^ 2 co x -\- /3 2 (co x — D, x ) \ 



On transformation, for purposes of comparison with the 

 electron theory, i. e. putting 



to^Xf + vA, 0»=X'/+i/A, . . . (50) 



we get 



7>i(X/-f-yA)=a 2 V 2 (A/ , + »'A)4-^ 2 {(^-^V+(^-^)A}, (51) 



and 



KX'/+ v'l) = /3 2 [(V - X)/+ (p' - A] - a 2 (X/+ v' A) 



-tW+^A). . . . (52) 



The equation (52) is the same as the equation (48) pro- 

 vided X' = 0, while from (51) we get (if X' = 0) an equation 

 of the form 



7+A = a'V/ + /3'Y+7' 2 A. 



Eemembering that/and A must vary as cos pt (say) we can 

 obviously adjust the constants and variables so as to put the 

 above equation in the form 



/+A = a''' 2 V 2 /, 



which is the second equation (38). [A and /differing in 

 value from the same quantities occurring in (51) and (52) 

 •each by a constant factor.] 



33. Kettler's equations are of the form (Glazebrook's 

 notation) 



mw + /AC'U = a 2 v 2 w 



i uC ? 7 + ^'U=-a 2 U-/3 2 //, &c. 



which yield as before 



m<o x 4- fi( y& x — a 2 v 2 ft> x 

 )uCa)* z + juIix= — a 2 ^— fi 2 £l x , &c, 

 which are the same equations as (51) and (52), if we put 

 mw x +■ fiC'n x -f+ A and pCa> x + pft* = =V+ (, A. 



