ACTING INVERSELY AS THE SQUARE OF THE DISTANCE. 159 
nitely small addition to it. For similar reasons we may also 
make 
av _ ules UE ty: ae ee aoe ay A 
dy 
and thus these Feber as well as V, will ees withinside ¢ 
determinate values varying continuously. The same will still 
hold good at the limit of 7. 
by 
de 
In regard to the differential coefficients of the higher orders, 
a different process must be adopted for points withinside ¢; for 
aes ads dX. 
example, it is not permissible to transform —— into 
dx 
pene 
edt. ; that is, into (oo —s ro) dt 
inasmuch as this sein strictly pen would be only 
a sign without any determinate clear signification. For, in fact, 
within every part of ¢, however small, which includes the point, 
portions may be determined, throughout which, if the integral 
be taken, it will exceed any given, positive, or negative value. 
Thus the essential condition fails, under which alone a definite 
signification might be attached to the integral, namely, the 
applicability of the method of exhaustion. 
8. 
Before undertaking this investigation in its generality, the 
consideration of a very simple particular case will be useful 
towards a clear understanding of the subject. 
Let ¢ be a sphere, cf which the semi-diameter = R, the cen- 
tre coinciding with the origin of coordinates; let the den- 
sity of the mass which fills the sphere be constant =k, and 
let us denote the distance of the point O from the centre by 
@= Vf (2?+y?+2*). It is well known that the potential has 
two different expressions, according as O is situated within or 
without the sphere. In the first case, 
2 2 
Ve ark R—— ake = 2akR?— awk (ae + y° +2"), 
and in the second case, 
42k R# 
Vv = 3p 
