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.

386 MR LARMOR, ON THE DYNAMICS OF A SYSTEM OF ELECTRONS OR IONS:
provided this is supposed to hold for every domain of integration, great or small, it will
follow that the electrons are the poles of a circuital or stream vector (f, g, h). If then
we write
fJ2 (dfd +d dh'\ dT- eI,
J kdx dy dz)
the variational equation will by Lagrange's method assume the form
8 f(T+ To- W+ n)dt=0
in which * is a function of position, initially undetermined but finally to be determined
so as to satisfy the above condition restricting the independence of the coordinates.
We have to vary this equation with respect to the displacement (f, g, h) belonging to
each element of the aether, supposed on our theory to be effectively at rest, and with
respect to the position (x, y, z) of each electron. All these variations being now treated
as independent, the coefficient of each of them rnust vanish, at all points of the aether and
for all electrons involved in it.
We now proceed to the variation. Bearing in mind that so far as regards aethereal
displacement
l i jFdir involves ffif r -d cdrT, that is iY/ifrriç1 T1-jT72,
because each pair of elements appear together twice ini the double integral of a product,
but only once in a double summation, we obtain as the terms involving f in the complete
variation
S dt Ffdr - 47rc2 dt fSfdr + dt Ij dfdr,
leading, through the usual integration by parts, to
f Fsfdr - dt FVfdT - 47T2 jdt fif d1r +\ dt fcfdIydz - fdt d Sfdr.
The coefficient of Sf must vanish in the volume integral, giving
dF dF P
4vr 'f-= _ d _t dx.........................................
Similar expressions hold for g and h. Again, the terms in the variation involving the
electron e at (x, y, z) are
S dte (xF + yG + zH) + mS (fdt (2 + y2 + Z2) - S fdte',
yielding as regards variation of the position of this electron
fdte (Fèc + GSy + Hsz + SF+y SG + z8H) + m dt (A&a + ySy + zi) -fdteST
in which Ux means the change of the velocity of the electron, so that we have on integration
by parts
F, ^ H7T d 'D F/2^ DG^ DH^\ 'dF dF dF
e FdSx + G3y + -dte DF Sx + - dG y + -DE z - x d x + d- y + d-z dz +...
+ ~m 3x + Yoy +z tz - m' Jdt ({x& + Y3y + zz)- dte (- x + dt y z+ d z
jdx dy dz