— 
158 GAUSS ON GENERAL PROPOSITIONS REGARDING FORCES 
case which occurs in nature, in which a determinate space is 
filled with matter of uniform or varying, but everywhere of 
finite, density. 
Let ¢ be the whole space containing the mass; d¢ an infini- 
tesimal element, of which the coordinates are a, 6, c, and the 
mass k dt; further, let V be the potential at the point O, the 
coordinates of which are 2, y,2; the distance of the element 
from the point O is therefore 
V ((a =a)? + (by)? +(e — 2) =F; 
consequently 
vast! 
a 
extended through the whole space ¢, which implies a triple in- 
tegration. It will be easily perceived that a real integration 
; ae iads 1 
may be executed, even if O is within the space, although =<) 
then becomes infinitely large for those elements which are infi- 
nitely near O. For if instead of a, b,c we introduce polar co- 
ordinates, by making 
a=x+rcosu, 6=y+rsinucosa, c=z+rsinusina, 
then dt becomes = 7 sinu.du.da.dr, and at the same 
time, 
Vafffkrsnu.du.dr.ar, 
where the integration, as it relates to 7, must be extended, from 
y = 0 to the values existing at the limit of ¢, from 4=0 to 
A= 2, and fromu =0 to w=. V will thus necessarily re- 
ceive a determinate finite value. 
Further, it is easy to perceive that we may here also say 
1 
NW: dr — pk(a—a) dt _ 
This expression, which, by the introduction of polar coordinates, 
becomes 
SL fk 008 usin udu. dr. dr, 
is susceptible of a real integration ; thus X receives a determi- 
nate finite value, which varies continuously, because all the ele- 
ments, situated infinitely near O, can only contribute an infi- 
