186 GAUSS ON GENERAL PROPOSITIONS REGARDING FORCES 
28. 
It is easy to convince ourselves that all the conclusions of the 
two last articles preserve their validity if S is not a closed surface, 
and if the masses are entirely contained in it only. Here the 
space T no longer exists ; all the points not in the surface itself 
belong to external infinite space; and if the potential have 
everywhere in the surface the constant value A, which differs 
from 0, it will have everywhere without the surface a smaller 
value with the same sign. 
What refers to the first case, A =0, obtains here also, but 
has no import, as in this case the potential V becomes= 0 in 
all points of space, so that SY is everywhere =0, ¢ signifying 
any straight line, whence it may easily be inferred, according to 
Art. 18, that the density in the surface is everywhere =0, so 
that the surface cannot contain any masses. 
Moreover, the above remark obtains generally, if the masses 
be contained only in the surface, even if it be a closed one, as it 
is evident that, according to the proposition in Art. 25, the value — 
of the potential in this case will be =0 in the whole interior 
space also. 
29. 
Before we proceed to the remaining investigations, in which 
masses, distributed continuously throughout a surface, form a 
principal feature, an important existing difference in their distri- 
bution must be attended to, by admitting either only masses 
having the same sign (which, for brevity, we will always con-— 
sider as positive), or masses having contrary signs also. Let a_ 
mass M be distributed over a surface in such manner, that to — 
each element of surface ds the mass mds may correspond, m — 
nine as as ae 
denoting, as heretofore, the density, and if mds extended over — 
Se 
the whole surface = M ; we shall call this a homogeneous distri- 
bution when m is everywhere positive, or at least nowhere ne-_ 
gative ; and a heteroyeneous distribution, when m is positive in’ 
some places and negative in others; so that M will be only the 
algebraic sum of the molecules of mass, or the absolute differ-_ 
ence of the positive and negative masses. A very special case of — 
heterogeneous distribution is where M=0, when it may ap-— 
pear improper to speak of the mass 0 being distributed over — 
the surface. 
opt. 
