188 GAUSS ON GENERAL PROPOSITIONS REGARDING FORCES 
2. That if parts of the surface are not so occupied, W must 
in those parts be greater, or at least cannot be less, than that q 
constant value. 
I. We must first demonstrate that if, instead of one mode of : 
distribution, another differing infinitely little from it be taken, 
by substituting m + » for m, the resulting variation of 0 will 
be expressed by 2 ff Weds. 
In fact, if we denote by © and 6 V, the variations of 0 and V, 
20 = f2V.mds+ f(V-2U) pas. 
But at the same time f V.mds = fv p ds, as may easily be 
seen from the theorem, Art-19, 8 V being no other than the 
potential of that mode of distribution in which » represents the 
density in each element of surface, so that what is here V, m, 3 V, 
#, may there be substituted for V, K, v, k, and ds may be taken 
both for dS and ds. Consequently 
10 =f @V—2U) pds = of Wuds, 
II. The variations » are obviously connected generally with a 
condition requiring that f » ds = 0; but in the present inves- 
tigation a second condition is also necessary; » must not be 
negative in the unoccupied portions of the surface, if such exist, 
for otherwise the distribution would cease to be homogeneous. 
III. Let us assume for a given distribution of M the exist- 
ence of unequal values of the quantity W at, different parts of — 
the surface. Let A be a quantity lying between the unequal — 
values of W; P the portion of the surface where the values of 
W are greater, and Q that where they are less than A ; further, 
let p, g be equal portions of the surface, the first belonging to 
P, and the second to @. This being supposed, let us give to 
the variation of m everywhere in p the constant negative value 
w= —yv, everywhere in g the positive value » = y, and in all the 
remaining portions of the surface the value 0. It is plain that 
the first condition in II. will thus be satisfied; but the second 
condition will still require that p shall have no unoccupied por- 
tions, which may always be effected, providing only that the 
whole of the portion P be not unoccupied. 
i 
| 
f 
