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. 399
so that, transposing,
s W = f fP + g'îQ + h'SR) dr,
in which the independent variable is now (P, Q, R).
On conducting the variation in the usual manner, and reducing from d;P/dt to S1'
by partial integration with respect to time (such as necessarily enters in the reduction
of the fundamental dynamical equation of Action) this leads to a relation of type, dF2 a3 dQ a2 dR
dP 47rC' dt 4vT2 dt'
where (a1, a2, a3)/4rc2 = (a - a, a - a-3, a, - a.,).
When the system is referred to its principal dielectric axes,
F - = K - 1 + K- 1 Q2 + KS- 1 R2'
87rC2 87rC2 87rC2
This analysis shows that rotational quality in the relation connecting (f', g', h')
and (P, Q, R) can come in through terms in the energy function that involve the
time-gradients: or, as may be shown in a similar manner, it may enter through terms
involving the space-gradients: but not otherwise. The latter terms introduce rotational
quality of the structural type, with which we are not now concerned. The former terms
lead to the magnetic type of rotation, here related to the vector (ai, a2, a3), which
must be determined by the impressed magnetic field or other exciting cause of vector
character: the existence of such mixed terms, involving (P, Q, R) and d/dt (P, Q, R),
in fact adds to the polarization a part at right angles to d/dt (P, Q, R) and to this
vector (ai, a3, a3), and equal to their vector product divided by 47rc2, which is in all cases
entirely of rotational character. Terms of the form of a quadratic function of the
gradients of (P, Q, R) by themselves would merely modify the form of the function
F2 so that its coefficients depend in part on the period of the vibration, that is, they
would be merged in optical dispersion of the ordinary type. The question also arises
whether the ordinary dielectric constants, namely the coefficients of the function F2 (P, Q, R),
are sensibly altered by an impressed magnetic field. This point can be settled by aid of
the principle of reversal. When the electric force and the impressed magnetic field and
the time are all reversed, the effect on the induced electric polarity must be simple
reversal: hence a reversal of the magnetic field cannot affect the coefficients in F2(P, Q, R):
hence these coefficients must depend on the square or other even power of the impressed
magnetic field: but the rotational terms depending on its first power are actually very
small, therefore any terms depending on its second power are wholly negligible. This is in
accord with Mascart's experimental result.
The right-hand sides of the equations of propagation in the material medium, as
above indicated, can thus, for light of period 27r/p, be expressed in the form?3-2 ( p -2 d2P dQ dR
p20- lp-2 dt -a3 dt + a dt
Kpd Q dR dP d2R dP dQ\
K2p- a - + a3 d p- a+t)'
dt3a1dt+ dt ' dtP