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.

356 PROF. LAMB, AN ELECTROMAGNETIC ILLUSTRATION OF THE THEORY
X =-k + d d - z U,
iZ=- = _ __yni - id d\
Z= knrn ---1 x -yd r! U j,
with a similar interpretation. The contour-lines of 'U,, are the lines of electric force,
and the lines of magnetic force are orthogonal to them.
5. The calculation of the energy carried outwards by the scattered waves leads to
some very simple results. By Poynting's theorem*, the rate at which the energy in
any given space is increasing is equal to the integral
4 1 {l (yY-/3Z) + rm (aZ- X) + n (3X- aY)} d.................. (52),
taken over the boundary of the space, 1, m, n denoting the direction-cosines of the
normal drawn inwards from the surface-element dS. The ambiguities which are known
to attend a partial use of this theorem will disappear if the space in question be that
included between a sphere of radius r, in the region of the scattered waves, and a
concentric sphere of radius so great that we may imagine it not to have been as yet
reached by the waves. The rate of propagation of energy outwards is therefore given
by the integral
-:| {(yZ-zY) +/3z(X-xZ) +(xT-yX)} dS................(53),
taken over the sphere of radius r.
Before applying this result, the values of a, /3, 7 and X, Y, Z must of course be
expressed ini real form. To take first a solution of the first type, since T,,, as given
by (31), will in general be complex, let us write
rnTl2 \ = j1 + i..................................... (54).
Restoring the time-factor in (48) and (49), and taking real parts, we find
= krl- (c -i - z dyJ {4 cos (at - kr + e) - < sin (aot - kr + ), &c., &c.,.........(55),
and
yZ- zY= - (y z - z d) [{, cos (o-t - kr + e) - sin (o-t - kr + e)}, &c., &c.,...... (56),
where e may be 0, or + T7r, or v, according to the value of n Hence the mean
value of the expression (53), per unit time, is found to be
c f[ Ç( dIn dbn'Y 2 ) d4nn dT, dÉPn dPt
Swkîîîri-2~ JJ d(k-d- dy d z+ zdx - x ~Jz) + xdx)
d dzy dy dx Xdz J +\ dy - dxp 3.
~* Phlj. Trans., 1884, p. 343.