MR. G. P. C. SEARLE ON PROBLEMS IN ELECTRIC CONVECTION. 
699 
charge has acquired an equilibrium distribution, it being supposed that there are no 
charges within the surface itself. Since the charge is in equilibrium the “ tendency 
to convection ” F must be everywhere perpendicular to the surface. This can only 
be when SP is constant all over the surface. From this I shall now show that, when 
the surface is closed, 'P is constant throughont the interior of the surface, and that 
consequently F is zero there. If n = 0 this last result implies that both E and H 
vanish also, as may be seen from equations (76) and (77). 
Let 'P = f (x, y, z) be the value of the “ electric convection potential ” at any point 
outside the charged surface S, and ^ = f (x , y, z) its value at any point inside S. 
The surface S is by supposition an equilibrium surface, so that T is constant at all 
points on it, and consequently f (x, y, z) — f (a?, y, z) = c, a constant, when x, y, z 
lies on S. If there are no charges in the interior of S then f satisfies the equation 
dy , d?r cr-f _ 
a d * "h j i “h TT — 0 
ax- ay- dz~ 
(78). 
Now, corresponding to the point x, y, z take in a new system of coordinates, the 
point £, y, £ such that 
g = x, y = y\/ cl, £ = zv/a. 
Then, corresponding to the surface S, we shall have a new surface S whose equation 
in terms of y, £ is 
<f> (£ y, £) =f(L y /\/£/\/«) = c. 
If we have also (/>' (£, y, £) = f (^, y/\/ a, £/v/ a), then the values of f and f at 
any point x, y , z are the same as those of <f> and (j.V at the point y, £. Now, since 
at all internal points f satisfies (78), it follows that at all points internal to S, </>' 
satisfies 
d^ 
. d 31 , M. 
drf dX~ 
= v 2 4 >' = o. 
Moreover ft is constant at all points on the surface 2. Hence </>' is the value 
of the electrostatic potential due to a distribution of electricity at rest, such that 
the surface t is an equipotential surface, and such that there are no charges within 
it. But in this case we know that (j.V is constant at all internal points. It follows, 
therefore, that f is constant at all points internal to the surface S. Hence, when a 
charged surface is in motion, and the charge has acquired an equilibrium distribution, 
the “ convection potential ” is constant throughout the interior of the surface. If 
there are no sources of magnetic disturbance in the field, so that n = 0, the constancy 
of 'P implies that both the electric and the magnetic forces vanish at all points internal 
to the charged surface. Thus if the only source of disturbance is the charged surface 
itself, the electric and magnetic forces due to it are entirely on the outside of the 
4 u 2 
