ACTING INVERSELY AS THE SQUARE OF THE DISTANCE. 189 
The result will be that 80 receives a negative value, as may easily 
be seen by putting this variation into the form 2 2 fiw —A) pds. 
Hence it is evident, that if, with a given distribution,—either 
unequal values of W obtain in the occupied portions of the 
surface,—or, with existing equality of the values in the occupied 
portions, lesser values are met with in the unoccupied portions,— 
a diminution of 0 may be obtained by a change in the distribu- 
tion, and that consequently with the minimum value the condi- 
tions enunciated in the above theorem must necessarily be ful- 
filled. 
32. 
If, for our special case (Art. 30), where U=0, so that W 
denotes simply the potential of the mass distributed through 
the surface, and © the integral /'V mds, we combine the theo- 
rem of the preceding article with that given in Art. 28, it fol- 
lows immediately, that with the minimum value of f” Vmds, 
the surface can have no unoccupied portions whatever ; for other- 
wise, even if the whole surface be a closed one, the occupied 
| portion must be considered as an unclosed surface, and in re- 
spect of it the unoccupied portion must be regarded as belonging 
to external space ; and in it, according to Art. 28, the potential 
must have a less value than in the occupied surface, whereas the 
theorem of the preceding article excludes a less value. 
It is thus demonstrated that there is a homogeneous distribu- 
tion of a given mass over the whole surface, in which no portion 
remains vacant, and from which there results an equal potential 
in all points of the surface. There is yet wanting to complete 
the demonstration of the theorem in Art. 30, to show that there 
can be only one mode of distribution which can satisfy these 
conditions ; the proof of this will appear in the sequel, as part 
of a more general theorem. 
That the minimum value of [ Vmds requires that there 
shall be no part of the surface unoccupied, may obviously be 
also expressed thus: for any distribution, in which a finite por- 
. 
tion of the surface remains vacant, the integral 4 Vmds re- 
a 
ceives a value which exceeds the minimum value bya finite 
difference. 
