Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

392 MR LARMOR, ON THE DYNAMICS OF A SYSTEM OF ELECTRONS OR IONS:
while Zeeman's measurements give e/m= 107: thus v2= ~c2e2/mr, so that, taking r to be
10-8, e= 10-21, -we obtain v = 10-3c; thus the orbital period cornes out just of the order
of the periods of ordinary light, which is an independent indication that the general
trend of this way of representing the phenomena is legitimate. With these orders of
magnitude, the ternis in the motional forces between two electrons are of orders ele2i/r,
ele2.c2/r2 as compared with their statical attraction of order 02ee2/r2 and the forces arising
from the impressed magnetic field H of order edHI; the ratios are thus of the order
of 10-6 to 1 to 3.10-9E. Thus when H exceeds 103, the forces of the impressed magnetic
field are more important than the motional forces between the ions; and in all cases
the effects arising from these two causes are so small that they can be taken as
independent and simply additive.
THE ZEEMAN EFFECT OF GYROSTATIC TYPE.
5. Sensible damping of the vibrations of the molecule owing to radiation cannot
actually cone into account, because the sharpness and fixity of position of the spectral lines
show that the vibrations subsist for a large number of periods without sensible change of
type. In fact it has been seen above that the motion of the system of electrons, on the
most general hypothesis, is determined by the principle of Action in the form
(T'- W) dt = 0
where T = m (2 + y2 + 2) + Y z y
x y z
I m n
thus it cones under the sarne class as the motion of a dynamical system involving latent
constant cyclic momenta, the Lagrangian function for such a system, as modified through
the elimination of the velocities corresponding to these nomenta by Routh, Kelvin, and
von Helmholtz, being of this type. The influence of the impressed magnetic field is thus
of the same character as that of gyrostatic quality imposed on a free system: and the
problem cornes under the general dynamical theory of the vibrations of cyclic systems*'. In
the special case above considered of massive positive ions, we can thus assert that the
motion relative to the moving axes is the same as the actual motion of the system with
its period altered through slight gyrostatic attachments to these positive ions. It is moreover known from the general theory of cyclic systems that each free period is either wholly
real or else a pure imaginary, whenever the unmodified system is stable so that its
potential energy is essentially positive: thus on no view can a magnetic field do anything
towards extinguishing or shortening the duration of the free vibrations of the molecule,
it only modifies their periods and introduces differences of phase between the various
coordinates into the principal modes of vibration of the system.
In the general case when ai is not the same for each ion in an independently vibrating
group in the molecule, the simple solution in terms of a bodily rotation fails, and it might
* Cf. Thomson and Tait, Nat. Phil., Ed. 2, Part I. pp. 370-416.