190 GAUSS ON GENERAL PROPOSITIONS REGARDING FORCES 
33. 
The leading feature in the mode of demonstration developed 
in the 31st Article, rests on the immediate recognition of the 
existence of a minimum value for ©, so long as we restrict our- 
selves to the homogeneous distributions of a given mass. If 
it were equally evident without this restriction, the conclusions 
in Art.31 would at once lead tothe result, that there is always, 
if not a homogeneous, yet a heterogeneous distribution of the given 
mass, for which W = V — U has at all points of the surface an 
equal value, as then the second condition (in Art. 31, II.) drops. 
But as the self-evidence is lost as soon as we dispense with the 
restriction to homogeneous distribution, we are compelled to 
seek for the rigorous demonstration of this, the most important 
proposition of our whole investigation, by a somewhat more 
artificial path : and the following appears the most simple which 
will conduct us to the end. 
Let us consider three different distributions of mass, in which, 
instead of the indefinite signs m for the density and V for the 
potential, we employ the following : 
Ll. m*h =m, Wie Ve 
IL. m=, Vie 
III. m= p V=w. 
I. is that homogeneous distribution of the positive mass M, 
for which /"V mds has a minimum value. 
II. is that homogeneous distribution of the same mass M, for — 
which ty —2-«U) mds has a minimum value, < signifying an — 
indeterminate constant coefficient. 
ane 
III. depends on I. and II., by making » = “, so that 
it is a heterogeneous distribution in which the total mass = 0. 
In Art. 31, V° was shown to be constant in the whole surface; 
V' — < U in the surface, so far as it is occupied in distribution 
II., and thence in the same portion of the surface vy — U be-— 
v' —v° 
<> 
Whether in the second distribution the whole surface be oc- 
cupied, or whether a greater or less portion remains unoccu-— 
pied, will depend on the coefficient «. As the second distri-— 
bution passes into the first if « = 0, so, generally, the portion of 
cause v= 
