710 
MR, G. F. C. SEARLE ON PROBLEMS IN ELECTRIC CONVECTION. 
Equivalent Distributions. 
21. The following simple proposition, now to be proved, I have found of great 
service when investigating the properties of a moving charged ellipsoid. 
Take any electrical system in motion and draw the series of equilibrium surfaces 
corresponding to successive values of MG Let My, MG, be the values of the “ convection 
potential ” corresponding to two of these surfaces, and suppose that the surface MG lies 
within MG,. Then if the same charge q be given to either of these two surfaces and 
be allowed to acquire its equilibrium distribution, then at all points not within the 
surface MG the effects of the two charged surfaces are identical. 
Fig. .3. 
Let A be any electrified surface in motion having a charge q with an equilibrium 
distribution, and suppose for the moment that this distribution is rigidly fixed. Let 
MG be the “ convection potential ” due to A at any point. Let B be any one of the 
equilibrium surfaces surrounding A. Now suppose that such a distribution is 
imparted to B, that at all points outside B there is no disturbance due to the pair of 
charged surfaces A and B. The electric force due to A and B therefore vanishes, and 
hence so also does the surface integral of normal electric displacement when taken 
over any surface enclosing both A and B. Any electric force due to O contributes 
nothing to this integral, since, as is seen from equations (26) to (28), it satisfies 
div E = 0 identically. The charge on B is therefore equal in amount and opposite 
in sign to that upon A, i.e., B has a charge — q. Let MG denote the convection 
potential due to this distribution on B. 
Now since there is no disturbance due to A and B outside B, it follows that M f has 
a constant value at all points outside B, and that, since M r vanishes at infinity, this 
constant value is zero. Now M' = MG + MG- But outside B, M' = 0. Hence 
outside B and at all points on B ’Lb = — MG- Now B was taken to be an equilibrium 
surface for A, so that MG is constant all over it. Lienee MG is also constant all over B, 
and therefore the distribution on B is the same as if B had been “ freely ” charged, 
except, of course, that the charge is now on the inner side of the surface B, whereas 
if B were freely charged it would be on the outer side of the surface. Since B has an 
