MR. G. F. C. SEARLE ON PROBLEMS IN ELECTRIC CONVECTION. 
701 
Hence, 
dy , d'Y dy 
— 4- — — = 0 
dz ^ dy dz 
or 
dy_ _ d^Y / dy 
dz dz / dy 
Thus the lines of magnetic force are given by the lines defined by y =z c, x = o', 
where c and c are variable parameters. 
Electrical Distribution on an Equilibrium Surface. 
17. The surface density at any point of a charged surface on which the electricity 
is in equilibrium is found from the fact that the surface-integral of the normal electric 
displacement taken over any closed surface is equal to the quantity of electricity 
within that surface. Hence, if E„ denote the electric force normal to the surface, 
and or the surface density, we have, just as in electrostatics, when there are no 
charges inside the surface, 
7 ~ EE ;t -- cr .. . . . . (79), 
since by § 16 the electric force vanishes inside the surface. 
The above statement refers to the case in which 12 = 0. 
When 12 does not vanish, we have instead, 
— K {E„ (outside) — E n (inside)} = cr .(80), 
where the electric forces are both reckoned in the same direction, i.e., along the 
outward drawn normal. 
Since no sources of magnetic disturbance reside on the surface, the part of E which 
is derived from fi is unchanged in passing through the surface, and the difference 
between the normal electric forces inside and outside may be computed from \P alone. 
Since y is constant throughout the interior of the surface, the part of E due to it 
vanishes inside the surface. 
Hence, if l, m, n be the direction cosines of the outward normal to the surface, we 
have by (26) to (28), 
whether VI2 vanish or not. 
Since F = — V'F and is normal to the surface we have 
<j — — 
K 
47T 
rry 
rj + 
ax a 
to dy n dy } 
dy a. dz J 
(81), 
