ACTING INVERSELY AS THE SQUARE OF THE DISTANCE. 169 
14, 
Before undertaking the investigation in its generality, it will be 
useful to consider a simple particular case. Let the surface be 
a portion, A, of the surface of a sphere, and let the density in it 
be uniform, or k be constant. Then V, X, are the values of the 
integral /” a Sf ae taken through A; and let V! 
X' represent the value of the same integral extended through 
the remainder of the spherical surface B, V°, X°, when extend- 
_ed through the whole of the surface of the sphere. Then V = 
v°— V', X° =X — X’. Further, let the radius of the sphere 
be called R; let the origin of coordinates be the centre of the 
sphere ; and let 4/ (z* + y? + *), or the distance of the point 
O from the centre of the sphere, be called = p. 
It is known that V? = 47k R, if Ois within the sphere ; and 
4akR?. 
_ if, on the other hand, O is without the sphere, V° = at 
the surface of the sphere the two values coincide. Hence the 
' : 5 ee: VO weie 
differential coefficient =— = 0 inside the sphere, and = — 
daz 
2 
ee Na * outside the sphere ; at the surface itself both values 
will hold good, each according to the sign of d w: these two 
yalues are equal, only if x = 0, which corresponds to Case IT. in 
the preceding Article. 
yo 
dex 
within and without the sphere, becomes on the surface a sign 
‘without any meaning, as a real integration is impossible, except 
in that particular case in which for elements lying infinitely 
near the surface, @ — x becomes an infinitesimal of a higher 
order than 7; namely, if y = 0, s = 0, y= + R; in whichcase 
the integration gives X° = 2 7k, thus not agreeing with either of 
0 
the values of A 
dx 
clear that this case belongs to Division I. in the foregoing Arti- 
cle. 
- If we now consider that when O is a point on the surface of 
The expression for X° having the same signification as 
, but rather with the mean of the two: it is 
I 
the sphere situated within A, X! and — have the same signi- 
fication, and that determinate continuously varying values, it is 
VOL. III. PART x. N 
