ACTING INVERSELY AS THE SQUARE OF THE DISTANCE. 191 
the surface remaining unoccupied for a determinate value of «, 
will diminish as < decreases, and will be quite filled up before 
¢=0. In particular cases a portion may remain unoccupied 
so long as « differs from 0, and has not changed its sign. It is 
sufficient for our purpose to take < as infinitely small; when it 
is easy to show that in any case no finite portion of surface can 
remain unoccupied ; for otherwise, according to the concluding 
remark in Art. 32, the integral if V' m’ ds rust exceed the inte- 
gral f'v° m° ds by a finite difference; if this be denoted by e, 
then the difference of the two integrals is 
Siwi=220) m! ds — fe — 260) mds =e 
—2ef U (m' —m’) ds, 
which for an infinitely small « preserves a positive value, in con- 
tradiction to the supposition that f” (V —2<¢U)mds has its 
minimum value in distribution II. 
Hence, we conclude, that if in the third distribution we take 
m — m° 
for » the limiting value of with an infinite decrease of 
e, then vy — U has a constant value in the whole surface. 
Ifnow we imagine a fourth distribution in which m = m°+ p, 
so that the whole mass = M, the resulting potential will be 
= V° + v; so that in the whole surface it will exceed U by the 
constant difference V° + v — U,whereby the theorem enunciated 
above is demonstrated. 
34. 
We have still to demonstrate that there is only one mode of 
distribution possible with a given mass M, in which V — U shall 
be constant throughout the whole surface. If there were two 
modes of distribution fulfilling this condition, then m and v being 
denoted by m! and v’ in the first, and by m" and v" in the second 
mode of distribution, then in a third mode, in which m is taken 
=m! —m'', the potential would be = V' — V", and consequently 
constant ; and the total mass would be = 0. According to Art. 
27, the potential must then necessarily = 0; and therefore, ac- 
