( 323 ) 
our formulae with too many indices, we shall omit them whenever 
this can be done without fear of confusion. 
The states of motion x, always occur in even number; in fact, 
corresponding to each state x 2+ , there will be a state whose 
frequency »(**+$ lies at the same distance from n 0 as n^~\ but on- 
the other side of it. The modes of motion jc 1 are conjugate two by 
two in the same way, with the exception, however, when k is odd, 
of one of them, which has the original frequency n 0 . 
The introduction of complex expressions, after the manner generally 
followed in problems of this kind, will be found very convenient. 
By this method one finds, for each mode of vibration, definite ratios 
between the quantities representing the components of the displace¬ 
ments of the several electrons from their positions of equilibrium. 
These ratios determine what may be called the “forms” of the 
vibrations, and it is especially to be noticed that, whereas in a particle 
subjected to the magnetic field only, the vibrations of the h modes 
mentioned above have unequal frequencies, a periodic external electric 
force can produce forced vibrations in these different modes, all 
taking place with the period of the force itself. 
\ 2. This case of an impressed electric force occurs when a ray 
of homogeneous light is propagated through the system, so that, if 
n is its frequency, the complex expressions for the dependent variables 
all contain the factor e int - While a particle'is vibrating, in all its 
different modes at the same time, under the influence of the alter¬ 
nating electric force existing in the beam of lightf it has an electric 
moment whose components are ? 
VV — ex, p y = ey, P- = ez, 
and the body is therefore the seat of an electric polarization (electric 
moment per unit of volume) for which we may write 
W = Np , 
where W is the number of particles in unit of volume. 
The equations of motion of a particle lead to the values 1 ) 
= Q 2 + (£* + &y) + W* ~ ^ 
% = Q 2+ («, - i<£ x ) + Qi- (<£y + »**). 
% = Qi 
the coefficients Q,, Q. 2 +, Q,^ indicating to what amount the vibrations 
in the modes x lf * s+ ,« a _ contribute to the polarization The first 
of these coefficients is given by 
l > Gf. Math. Encykl. Y 22, § 47. 
