ACTING INVERSELY AS THE SQUARE OF THE DISTANCE. 155 
agent and the recipient. ‘These components are represented by 
the partial differential coefficients 
af af a= 
r r r 
bai ud Piduw des 
If therefore several agents »°, u', uw’, &c. act on the same point 
O, from the distances r°, 7’, r", &c., and if we write 
0 I 
+545 + & =2S=V, 
then the components of the whole force acting at O will be 
represented by 
edV edV edV 
If the agents do not act from discoatinuous points, but from 
a continuous line, surface, or solid, then, instead of the summa- 
tion =, we have a single, a double, or a triple integration. The 
last case is the only one which exists in nature; but as it is 
often possible, under certain limitations, to substitute forces 
imagined to be concentrated in points continuously distributed 
on lines or surfaces, we will include these cases in our investi- 
gation, and we shall not scruple to speak of masses distributed 
over a surface, or upon a line, or concentrated in a point ; inas- 
much as the expression “mass” will here mean merely the source 
from which the attracting or repelling forces are imagined to 
proceed. 
oe 
If, then, we denote by 2, y, z the rectangular coordinates of 
any point in space, and by V the sum of all the active particles 
of the mass, divided respectively by their distances from that 
point, so that, according to the condition of the investigation 
on each occasion, the negative particles may be either excluded 
or suitably treated, V becomes a function of x,y,z; and the 
investigation of the properties of this function will be itself the 
key to the theory of the attracting or repelling forces. For 
convenience I will give to the function V a special denomination, 
calling it the potential of the masses to which it relates. This re- 
stricted conception of the potential suffices for our present in- 
vestigation; in the case of other laws of attraction, than those 
which act in the inverse ratio of the square of the distance, or 
for that of the fourth case mentioned in Art. 1, we may, in a 
M 2 
