THE ELECTRIC AND LUMINIFEROUS MEDIUM. 
237 
of radiation ac;ross a medium permeated by molecules, each consisting of a system of 
electrons in steady orbital motion, and each capable of free oscillations about the 
steady state of motion with definite free periods analogous to those of the planetary 
inequalities of the Solar System; and Its analysis will in fact resemble Laplace’s 
general investigation of the latter problem. If 6^, 9^, . . . represent small deviations 
from the state of steady motion of a molecule, so that the coordinates of the system 
(0 + ^vf-z {t) + • • -5 the kinetic and potential energy of the molecule when 
expanded in powers of these small quantities will assume the forms 
T = const. + [ 0 ^, 9 ,,. . .]i + [ 0 „ 9 ,,. . .]i + [ 9 „ 9 ,,. ..], -f [ 9 „ 9 , 
+ [{9,, 9„ . ..} {9„ 0„ . ..}] 
W = const. + [ 9 ^, 9 . 2 ,. . .]i + [ 9 ^, 9 . 2 ,. . .Jg, 
2) 
1 
•J2 
where the terms in T and denote functions of the various degrees of these 
velocities and displacements, the last term in T being a lineo-linear function of them 
jointly. From these expressions the free vibrations are determined by the Lagrancuan 
method. As the undisturbed motion is steady, the type of a free vibration must be 
the same at whatever time it is excited, therefore the coefficients in T and W are all 
independent of the time ; indeed if they were not constant the system could have no 
free periodic vibrations at all. The equations of the steady motion show that there 
can be no terms in T — W of the first degree in the displacements, when the 
coordinates are propeily chosen.'^ At the present stage we may conveniently by 
transformation of coordinates express the Lagrangian function, on which the motion 
in the molecule depends, in the form 
1 W — [02, 02,... . 9,^1 + \ {A^^p + A30p + ....+ A„^'p] 
— ^ {Cti9i + + ....+ a„9,Pj -|- + .... + + ^ 21 ^ 2^1 + ••••, 
from which, by a property which is an Immediate corollary of the Action principle, 
we may subtract any perfect difierential coefficient with respect to time, for exam])le 
here 
[^n ^35 • • • • ^/Jl 4" d/dt {^l>ii9i ^(^22 ^2l) ^1^2 4" • • • ■}’ 
without affecting the course of the motion, leaving thus an effective Leujrangian 
function 
L = i {A2^2^ 4- 4--4- i 4- cifi 4-_4- 
2 - 029 ^:)+ ....}. 
24. We have now to determine the vibrations forced on this molecule by the 
uniform alternating field of electric force, say P parallel to the x axis, belonging to 
the radiation which is traversing the medium. Bearing in mind that the wave 
length covers about 10'^ molecules, it appears that if/" denote the total intensity 
* This analysis, so far, is as given by Rolxh, “Advanced Rigid Dyn.aniics,” § III. 
