ACTING INVERSELY AS THE SQUARE OF THE DISTANCE. 167 
that point be situated in the separating surface of two spaces, in 
each of which, taken apart, the density varies continuously, but 
_yaries discontinuously in passing from one to the other, then, 
2 2 2 
generally, — TP ; ae have each there two different 
values, and what has been said at the close of Art. 8. is true of 
the aggregate of those quantities. 
12. 
We have included in our investigation, as has been already 
remarked, the ideal case in which the attracting or repelling 
forces are supposed to proceed from the parts of a surface, and 
in doing so have permitted ourselves to represent the acting 
mass as distributed over the surface. By density at any point 
of the surface we understand in this case the quotient of the 
mass contained in a surface element to which the point be- 
longs, divided by that element. This density may be uniform 
in all the points of the surface, or not uniform, and in the 
latter case it may either vary in the whole surface continuously 
(1. e. differing infinitely little between any two points infinitely 
near to each other), or the whole surface may be resolved into 
two or more portions, in each of which the variation is conti- 
nuous, whilst, in passing from one to another, the change is 
sudden. We may, moreover, imagine such a distribution, that, 
notwithstanding the whole mass be finite, the density in par- 
ticular points or lines shall be infinitely great. The surface 
itself not being a plane, will generally have a continuous curva- 
ture, without, however, excluding an interruption in singular 
points (cusps) or lines (edges). 
These suppositions being made, the potential receives at each 
point of the surface, where the density is not infinitely great, a 
determinate finite value, between which and the value at a se- 
cond point infinitely near to it, either in the surface or without 
it, there can be only an infinitely small difference*; or, in 
other words, in any line, whether in the surface or cutting it, 
the potential varies continuously. 
* It is easy to convince oneself of the finite value of the integral which ex- 
presses the potential, by resolving the surface into elements in a manner 
similar to that in which it is done in Art. 15., and it will at once be seen from 
thence that the parts of the surface infinitely near to the two points in question, 
contribute infinitely little to the whole integral; from whence what is said 
above may be easily demonstrated. 
