MAGNUSSON — ANOMALOUS DISPERSION OP CTANIN. 263 



To derive expressions for these forces, Helmholtz made the 

 following assumptions : 



1st. The impressed forces are supposed to vanish; hence: 



X 1 =X 1 ' = 0. 



2nd. The medium is supposed to be perfectly elastic ; hence : 



d 2 u 

 x = a2 d^~ < 17a )- 



dU 

 X 1 =-a*U-r 2 - d f (17b). 



3rd. The force of restitution in an elastic medium is propor- 

 tional to the displacement ; hence : 



A = /?» (U — u) (18). 



4th. The action is supposed to be confined to the element of 

 volume dv; hence: 



A + A 1= (19.) 



Substituting the values thus found for X, X„ X', X' l5 A and 

 Aj in (15a) and (lob) we have : 



d 2 u d*u 



m ~dt»~ = a * dz»~ + £" lU — u) (20b). 



d 8 U dU 



m ' dF" = ~ /Sa (U ~ u) ~ a * U ~~ r * dF (2° b )« 



These are the fundamental differential equations in Helm- 

 holtz's dispersion theory. Solving, he finds: 



B 



^_ [7i-A)2i_i] 



. 4 m L^ m' / A 2 m J 



, v , ., BA 2 mm' X* m 

 U ■ ~~ X ~ 1 - Jnh»; + 77 B \ A* TV . „ A» (21a). 



- [O-^^-^' + o- 



A* 



B« G A* 



m m' A s m 

 2 Mo X = p ^1 TF (21b). 



Where ^ equals the refractive index at perpendicular inci- 

 dence ; x , the extinction coefficient, for X at perpendicular in- 

 cidence ; and A m) wave length of absorption band. 



