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.

404 MR LARMOR, ON THE DYNAMICS 0F A SYSTEM OF ELECTRONS OR IONS:
so that X=r = BrCOs (qrt + /3y), yr = - ZBr sin (q.t + /r),
which represents similarly a series.of left-handed circular vibrations. The vibrations of z
type will of course be linear in form.
Thus supposing the effective masses and charges of the various ions to be entirely
arbitrary, the effect of an impressed magnetic field will be to triple the periods and
polarize the constituents in the Zeeman manner, provided the potential energy of the
mutual forces of the ions is any quadratic function of the coordinates of the vibrations
which satisfies the condition of being invariant in form with respect to rotation of the
axes of coordinates around the axis of the magnetic field.
The essential difference between the type of this system and that of the one
previously considered will appear when the latter is derived on the lines of the present
procedure. The equations are... 2k dW
}- + lK,'__ _2k dW
rï - r] —,IC,
m d '
k dW
m dz
On writing:'= e'Kt, '= e- Ktr,
2.,. 2k dW
they become ' + c - - 2 dW
m d''
k 2 dTWI
m dz
The form W will be unaltered when it is expressed in terms of:', V', provided
it depends only on the mutual configuration of the ions, and K is the same for all of
them; hence when KA2 is negligible compared with unity, (:', V', z) are determined by
the sane equations as would give (e, V7, z) on the absence of a magnetic field: and
from this the previous results follow.
10. We have thus reached the following position. Let the coordinates (x, y, z)
of an ion be resolved into two parts, namely (x1, yt, zj) which are known functions of
the time and represent its mean or steady motion, and (x', y', z') which are the small
disturbance of the steady motion constituting the optical vibrations. When this substitution is made in the dynamical equations the quantities relating to the steady motion
should cancel each other, as usual; and there will remain equations, of the original
form, involving (x', y', z') from which the accents may now be removed. The forces
relating to these new coordinates will still be derivable from a potential energy function: