ACTING INVERSELY AS THE SQUARE OF THE DISTANCE. 187 
30. 
It is self-evident, that however a mass M may be distri- 
buted homogeneously over a surface, the positive potential V 
everywhere resulting will, at every point of the surface, be greater 
than 37 denoting the greatest distance between any two points 
of the surface; the potential could only have this value at one 
extremity of the line 7, if the whole mass were concentrated in 
the point at the other extremity, a case which cannot come into 
question here, as we are treating solely of continuous distribu- 
tion, where to each element of surface ds only an infinitely 
small mass mds corresponds. The integral f Vmds, extended 
over the whole surface, is thus in every case greater than 
M? : 
of 3 mds, or Sg Mat that there must necessarily be one mode 
of homogeneous distribution for which that integral has a mi- 
nimum value. It may here be premised, that one of the objects 
of the following investigations is to demonstrate that with such 
a distribution, in which Jf V mds has its minimum vaiue, the 
potential V will have one and the same value at every point of 
the surface, that no parts of the surface can remain vacant, and 
that there is only one such mode of distribution. For brevity, 
however, the investigation will be conducted from the com- 
mencement in a more comprehensive form. 
31. 
Let U denote a’quantity, having at every point of the surface 
a given finite value varying continuously. The integral 
a=f (V—2U) mds 
extended over the whole surface may, it is true, have very unequal 
values, according to the diversity of the homogeneous distribu- 
tion of the mass M ; but it is evident that for one such mode of 
distribution, there must be a minimum value of this integral. 
We propose to demonstrate the 
THEOREM, that for such a mode of distribution 
1. The difference V—U=W will have a constant value at 
every part of the surface which is occupied by parts of M. 
o 2 
