ACTING INVERSELY AS THE SQUARE OF THE DISTANCE. 179 
22. 
THEOREM.—If ds be the element of a surface bounding a 
coherent finite space, P the force which masses, distributed in 
any manner, exert at ds in a direction normal to the surface, a 
force directed towards the interior or towards the exterior be- 
ing considered as positive, according as attracting or repelling 
forces are deemed positive; then the integral “yi P ds, extend- 
ed through the whole surface = 427M +22M’, M denoting 
the aggregate of the masses in the interior, and M! that of those 
on the surface distributed continuously. 
Demonstration.—If we denote by U dp that part of P which 
is derived from the element of mass dy, by r the distance of 
the element dy. from ds, and by u the angle which the normal 
directed towards the interior makes with r at ds, then U 
= = “. But with respect to any given dp, by virtue of a 
theorem demonstrated in Art. VI. of the Theoria Attractionis 
=" ad3= 0, 25, or 
47m, according as dp is without the space bounded by the sur- 
face, in the surface itself, or within the space in question. Now, 
as sf P ds is equivalent to the amount of all the dp. af Uds, 
our theorem follows immediately. 
With respect to the auxiliary proposition which has been 
used here, it must be remarked, that in the form in which it has 
been enunciated in the work referred to, a modification is neces- 
sary for a particular case. The distance of a given point from 
the element ds being denoted by r, in case the point should be 
Corporum spheroidicorum ellipticorum, f- 
in the surface itself, the formula /” oe ds=2- is true only 
when the continuity of the curvature of the surface is uninter- 
rupted at the point. But such an interruption occurs if the 
point be situated in an edge or in a cusp, and, in such case, for 
2m we must substitute the area of the figure which the system 
of straight lines, touching the surface at the point in question, 
cuts off from a spherical surface described with radius = 1 about 
that point as centre. 
But as such exceptional cases respect only lines or points, and 
thus not parts of the surface, but only boundaries between parts, 
