ACTING INVERSELY AS THE SQUARE OF THE DISTANCE. 185 
would be constant in all the space within the sphere, and hence, 
according to Art. 21, it would also be constant in all the infinite 
space T’, and in both=0, in contradiction to the supposition 
that at the limit of this space at the surface S it is different 
from 0, and the impossibility of its changing discontinuously 
from thence. The second supposition, on the other hand, would 
be in contradiction with that demonstrated under I., if B were 
either = 0 or=A, thence B must necessarily fall between O 
and A. 
27. 
THEOREM.—In thetheorem of the preceding article,the first 
case, or the zero value of the constant potential A, can only ob- 
tain if the sum of all the masses is itself =0, and the second 
only if this sum is not =0. 
Demonstration —Let ds be the element of surface of any 
sphere enclosing the space T, R its radius, M the sum of all the 
masses, and V their potential at ds. As according to the 
theorem in Art. 20 the integral if Vds=42zRM;; but ac- 
cording to the theorem immediately preceding, in the first case, 
or for A =0, the potential V at all points of the spherical surface 
will be =0; and in the second case, on the other hand, it will 
be less than A, and will have the same sign; then in the first 
case 47 R M=0O, so that M=0; in the second case 427 RM 
and therefore M likewise must have the same sign as A. It 
is evident that in the second case 47 RM will be less than 
; St Mor 4x R? A, and M will be léss than RA, or A will be 
‘| greater than = 
The second part of this theorem, in combination with the one 
in the preceding article, may evidently be also expressed in the 
.| following manner :— 
| If the algebraical sum of masses, contained in a space bound- 
| ed by a closed surface, or being also in part distributed conti- 
| nuously in the surface itself, be =0, and their potential have a 
| constant value in all points of the surface, this value will neces- 
sarily be=0, and will at the same time apply to all external 
, | Space, where consequently the action of the forces exerted by 
these masses completely destroy each other. 
VOL. IL]. PART x. ) 
