319 
is only possible by caleulating the value of P X IE X and of i x jE x \ 
and this cannot be done unless some special assumptions as to the 
motion of the electrons contained in the body are adopted. 
§ 4. In the present paper the conduction current (due to free 
electrons) will be left out of account; consequently, all we have to 
do is to find the value of the ratio P x /E x . This we now proceed 
to calculate. We shall begin by writing down the equations of mo¬ 
tion of an electron vibrating in a molecule under the action of 
external and internal forces. 
Let e be the electric charge of an electron belonging to a defi¬ 
nite class or category and let N represent the number of electrons 
of the class contemplated, per unit volume of the medium. We have, 
writing rj , f for the components of the displacement of the elec¬ 
tron (reckoned from its position of equilibrium): 
P x = 2eÇN P y = 2erjN P K = 2e£N- (1) 
the summation is extended to unit volume of the medium and has 
to include all classes of electrons present. 
Let us write as the equation of motion in the direction of x: 
1 + 8*1 + VI =£(*1 + < 3 P „). ( 2 ) 
Here m represents the effective mass of the electron, n 0 the fre¬ 
quency of the free or "natural” vibration, k is a constant coefficient 
and co a numerical factor whose value has been discussed by H. 
A. Lorentz. He finds*■) that cD has the value | n in some simple 
cases and considers that in general it will not differ much from 
Consequently, if we write 
ù = in(l-\-o) (3) 
G will be small in comparison to unity. On the other hand, it must 
not be forgotten that the quantity a will depend on the density of 
the body and also on the nature of the restitutive force acting on 
the electron i.e. on the value of n 0 2 ). 
It is doubtful how far we are at liberty to consider equation 
q La théorie électromagnétique de Maxwell et son applica¬ 
tion aux corps mouvants, Leiden, 1892, §§ 105 et 106. 
2 ) La théorie, § 106. 
5 * 
