ON OPTICAL THEORIES. 259 
Let 6 be the angle between this direction and that to the point at which 
the disturbance is required, p the distance to the point, and a the angle 
between the plane zOp and some fixed plane. 
Let 6’, 6” denote disturbances perpendicular to the radius vector in 
the plane «Op, 
P’ P” along the radius vector, and 
N’ N” normal to the wave plane «Op. 
Then it follows that if we have small electric displacements X/e~*8-Vo 
parallel to z throughout the small sphere (47R? = dv), that 
om — 20, OE sinter oer) 
0 7p 
mh (33) 
P= oc ye cos Be~® &- V9 dy | 
WN’ = 0” =P” =0 
N” sat Glen 8d \ Lies Ui) Vide 9 Bull brea 
where C,=C, (2 xt x) 
2 ‘ 
4 =0, (1- #8) ate 
tine 0 bp bp? 
This agrees with the results given by Stokes and Lord Rayleigh, already 
quoted,’ N” being proportional to the rotation. The effect of a general 
arbitrary electric and magnetic displacement is then found, 
In considering the optical problem it is pointed out that electric 
displacement is always accompanied by magnetic, and that the effects of 
the two must be considered, and according to the views of Professor 
Rowland the two must be considered independently. From the relation 
between the electrostatic and electromagnetic energy, it follows that if 
there be an electric displacement X’eY there will be a magnetic Ye®¥ 
where 
y" =_ dr yy 
VK 
The electric displacement at any other point of space is found and 
expressed as below. Let the origin be the point at which X’, Y” act ; the 
axis of z the normal to the plane of X/aY”; p the direction in which the 
effect—at a point A—is required; the angle zO.A = 0, and the angle between 
#OA and zOz = ¢; and let P’, 0’, ® be the displacements along OA, and 
normal to 6 and 9. 
Then 
3b2X'V } 0 : 7 
a vow oe ¢ [a + cos 8) (1 =) - er | en ib -Vo) 
3U2X'V . a 1 : 
qd = — EET A \ ek —ib(p-Vd) . 
Bra sin [ (1 + cos 0) (1 x) PP e (36) 
3ibX'V . iq_; 
p’ = pea — ib (ep — Vt) 
dap? sin 8 cos p [ 1 ae 
1 See p. 201. 
