THE ELECTRIC AND TjUMINIFEROUS MEDIUM. 
239 
the molecules. By the Lagrangian method, these equations expressed for a 
molecule of the first kind and for radiation of the above type , form a system, skew 
symmetric in so far as of t 3 q:)e 
4- «i) + .... 4 e,,.cie,, - cqPi 4 cVK^c-^r/P = 0, 
wherein 
Ci^l 4 4 • • • • 4 — f\ ~ 0. 
They give the relation 
e,„ci, - Cl] Pi 4 K'=c V {Ai<z'^ 4 ff], c-x.q, .c,.?, c\] Pj 
in which the denominator represents a skew determinant, and each of the two 
coefficients in the numerator the same determinant bordered. The denominator 
involves when expanded only even pov/ers of q, and when equated to zero it gives 
the periods of the free vibrations in the molecule ; as these are all real the roots in 
q^ must be all real and negative. The second term in the numerator has c“^ as a 
factor; we may therefore neglect it as has been done in the previous pajoer; this 
means that the elasticity of the rether is so high compared with its inertia that the 
pull exerted by it on the molecule will be important while the interaction of its 
kinetic energy will be negligible. The remaining determinant in the numerator, 
when expanded, contains only even powers of q and is of order lower by two than 
the denominator. Hence writing — for i/, so that 2ttIp is the period of the radia¬ 
tion, and expanding in partial fractions, we can express the equation in the form 
47rC 
2/1 _ 0 \ 
3 "t" 
'Pl-f 
, 4 • • . • 4 
On 
Vn - r 
in which g. 2 , . . . , g,i are real quantities, j)ositive or negative. 
Now the index of refraction p or K'“ of the compound medium is given l>y the 
formula(K'- l)/(K' 42 )=:R,.| 7 rcyi/P, 4 n 2 -l^cy 3 /P, 4 . . . . ' 
The final result is thus 
K^- 1 _ 
K' + 2 ~ 
'Enm, where m = - 
0 i 
4 
02 
Pi- - Pi - p 
,0 + 
+ 
On 
Pn - r 
SO that it is m and not that has an infinity at each free period of the molecule. 
We here again arrive at Lorentz’s refraction-equivalent, and the theorem that it is 
an additive physical constant; but with the important addition that it is the law of 
dispersion of the molecular refraction-equivalent m, equal to {p^ — 1) / (p^ 4 2) p, of 
each constituent of the medium, not that of the refractive index of the aggregate, 
which admits of simple theoretical expression. In physical investigations concerning 
laws of dispersion, it is thus essential to deal with simple substances ; the dispersion 
m the molecular refraction-constant of a mixture, and no doubt also to some extent 
