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. 405
and as by hypothesis the vibrations are all 'cycloidal' or simple harmonie, this function
must be homogeneous and quadratic in these coordinates. The total potential energy
must be determined by the instantaneous configuration of the system, and will therefore
remain of the same form when referred to new axes of coordinates. This confines the
quadratic part representing the energy of the disturbance to the form given above:
the vibration of each ion will then in general consist of a system of elliptic oscillations
of all the various free periods, equal in number to the ions: and the effect of an
impressed magnetic field will be to triple each vibration-period and to polarize the
constituents in the Zeeman manner. The steady or constitutive motion of the system
must be so adjusted that it does not sensibly radiate: otherwise it would gradually
alter by loss of its energy.
As the axis of the magnetic field may be any axis in the molecule, the function
which represents the potential energy must thus be such that the vibrations resolved
parallel to any axis form an independent system: hence it is confined to the form
W = - _,, (X,.r'- Xs)2 + (y1' - y1)2 + (z. ' - Z,' /)2} +,Brs (x,'Xs, + yr'yS + Z/Zs),
- Ars {(Rt' - es') (]r' - r7s) + (Zr - Zs')2} + IB1 (' s'4 + - s's'/ + 22v/Zs').
Thus in the absence of a magnetic field the vibrations of the x coordinates, of the
y coordinates, and of the z coordinates of the ions will form independent systems of
precisely similar character. It is in fact only under this condition that it is possible
for the components, parallel to any plane, of the elliptic harmonic vibrational types of
the various ions, to form a system of circular vibrations with common sense of rotation.
If n/k = X and mrc/k= X', the equations of motion are of type
dW dW
' drV d^ d
The periods of the right-handed circular vibrations, of type xc eLpt, period 27r/p, will be
given by the equation
- p-X2 - X'p -A1,, 1 C,3, C14,... Cn = 0,
C21, - x2p2-\p-x An,., 23, C24,... C0
C31, c3, - X2 - X3p - A, C34,... C3.........................................................................................................
in which Cs= A,. + B,,: those of the left-handed circular vibrations by changing the
sign of each X' in this equation: those of the plane-polarized vibrations, which are the
natural periods of the molecule, by making X' null. On account of the great number
of the constants, compared with the number of free periods, simple relations among the
periods can only arise from limitations of the generality of the system.
The duplication or triplication observed in the constituent Zeeman lines would on
this theory arise from the presence of two or three equal roots in the period equation