‘ACTING INVERSELY AS THE SQUARE OF THE DISTANCE. 181 
_ Hence, in any distribution of the masses, the equation 
dv 
dp 
denotes the masses contained in the internal space, it being well 
understood that even if there be on the surface itself masses dis- 
tributed continuously, they must be regarded as belonging to 
those in the interior, or excluded from them, according as the 
values which apply to external or to internal space have been 
ds= 472M is universally true in the sense that M 
chosen for Wyk 
Consequently, if there are no masses in the internal space, 
; 
then, if we understand in every case by > the values belong- 
: Sa en aV 
ing to the interior, uif ae ads = 0. 
24. 
Under the same suppositions as those at the close of the last 
article, and denoting by T the space in question, and by g the 
whole resultant force at the element dT of the masses, either 
outside the space, or distributed continuously in the surface, we 
have the following important 
THEOREM : SVZas= — feat, 
if the first integral be extended through the whole surface, and 
the second throughout the whole space T. 
Demonsiration.—Introducing rectangular coordinates 2, y, z, 
let us consider in the first place a straight line, cutting the space 
T parallel to the axis of 2, so that y and z have constant values. 
From the identical equation 
a4. aV HbR eV 
a (¥ az) = wre! bea Sores 
it follows that the integral 
f(a ye — de: 
being extended throughout that portion of the straight line 
which falls within T will be equal to the difference between the 
dV : ; 
two values of V ia at the extreme points, inasmuch as the 
straight line cuts the bounding surface only twice ; or, generally, 
. * . y . 
it will be = >eV a putting for V x the respective values 
