ACTING INVERSELY AS THE SQUARE OF THE DISTANCE. 177 
the limiting values of those which relate to two separate sur- 
faces, by causing their distance from each other to decrease in- 
finitely ; for which purpose it is only necessary to assume these 
two surfaces to be equal and parallel with each other. It is true 
that this kind of demonstration is immediately applicable only 
when the surface is such that the normals at all its points form 
acute angles with a determined straight line. A surface, in which 
this condition fails (as is always the case with a closed surface), 
must first be decomposed into two or more parts, which, taken 
singly, satisfy that condition, so that it is easy to reduce this case 
to the preceding one. 
20. 
If we apply the theorem of the preceding article to the case 
when the second system of masses, with a uniform density k= 1, 
is distributed over the surface of a sphere, the radius of which 
= R, then the potential v arising therefrom is constant with- 
in the sphere and = 47R; at any point without the sphere, of 
2 
which the distance from the centre is r, v = , or it is just 
= 
as great as the potential of a mass 47 R*, placed at that point, 
is at the centre; on the surface of the sphere the two values of v 
coincide. Thus, if the first system of masses be altogether within 
the sphere, then = M v will be equal to the product of the whole 
mass of this system multiplied into 4R; but if this system of 
masses be altogether without the sphere, then = M » will be equal 
to the product of the potential of this mass at the centre of the 
sphere multiplied into 4 z R?; lastly, if the first system of masses 
is distributed continuously on the surface of the sphere, then for 
: pe ku ds the two expressions give the same result. Hence fol- 
lows the 
THEOREM.—If V denote the potential of a mass, howso- 
ever distributed, in the element ds ofthe surface of a sphere de- 
scribed with the radius R, then if we integrate through the whole 
surface of the sphere 
SVds=4e (R M° + R2V°), 
if we denote by M° the whole of the mass within the sphere, by 
V° the potential at the centre of the sphere of the external mass, 
and class at pleasure with the internal, or with the external 
masses, the masses which may be distributed continuously upon 
the surface of the sphere. 
