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.

AND THE INFLUENCE OF A MAGNETIC FIELD ON OPTICAL PHENOMENA. 403
so that 2x= + 77, 2ty= —,
d d d d d d
and therefore 2 = -_ -, 2- =d +- d;
da dx dy dc dx dy
the equations become
m (+ + u4)=X LY,
m (r - )= X - Y,
mz =Z.
If therefore X + Y is a function only of the e coordinates of the electrons, and X -Y
a function only of the q7 coordinates, and Z only of the z coordinates, these groups of
coordinates will be determined from three independent systems of equations.
On our hypothesis of ions moving with velocities of an order below that of radiation,
the mutual forces acting on them are derived from a potential energy function: thus
(X, Y, Z)=-k (d d d W
dx' dy' dz
where k may be supposed to vary from one ion to another, being equal to the electric
charge when the mutual forces are considered to be wholly of electric origin. Then
1 2k dTW 1i 21c dW
(X+ tY) =, - -(X -Y)...
m mv dr ' m m d:
The solution of the coinplete system of equations, three for each ion, will in any case
involve the expression of e, r7, z for each ion as a sum of harmonic terms of the form eLPt
each with a complex numerical coefficient; but when the coefficients of one of them are
assigned those of the others are determined. The vibration for each ion is thus compounded
of a systein of elliptic harmonic motions of definite forms and phases. Their components in
the plane ~, 7 will be circular vibrations only when the t and V coordinates vary independently of each other, that is when dW/drq is a function of the t coordinates of the ions
alone and dW/de a function of the q coordinates alone. This condition can only be satisfied,
W being real, when it is a linear function of z2 and of products of the form g.r, or rs:
it may thus be any quadratic function of the coordinates which is invariant in forni as
regards rotation of the axes of x, y around the axis of z. Under these circumstances the
free periods for e coordinates, 77 coordinates, and z coordinates will all be independent, and
either real or pure imaginary*: in an actual molecule they will be real. For example a
permanent vibration of | type will be represented by:r = >AreLPt+Lar
ar being chosen so that Ar is real: thus. = Ar cos (p,.t + ar), y, = SA, sin (pt + ar)
representing a series of right-handed circular vibrations, each series having definite phases
and also amplitudes in definite ratios for the various ions. Again for the 7 type we have
7lr = CBr.e''t+s t,
* Routh, Essay on Stability, 1887, p. 78; Dynamics, vol. n., ~ 319.
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