( 260 5 
bouring points, it may come to pass that there are a certain number 
of intersections with one plane ds and none at all with another 
plane of the same direction. We shall meet this difficulty by irregu- 
larly undulating the element of surface, in such a way that the 
distances of its points from a plane do are of the same order of mag- 
nitude as the distance d, and that the direction of the normal is very 
near that of the normal # to this plane; so that the extent of the 
element and the normal to it may still be denoted by do and n. It 
is clear that, if MN is a constant, the number of intersections of the 
vectors Q FR with such an undulated element may again be said to 
depend only on its direction and magnitude, and that it may still 
be represented by the formula (10). 
The same formula will hold in case the value of WV should slowly 
change from point to point, provided we take for V the value belong- 
ing to the centre of gravity of the element. 
hb. Let us apply the above result to the elements do of a closed 
surface 6. Let n, be the number of ends #, and n, the number of 
starting points Q lying within o. 
Supposing the normal # to be drawn in the outward direction, 
we may write for the difference of these numbers 
| ING BG. 5 vide ene SEN 
an expression, which of course can only be different from O, if MN 
changes from point to point. 
c. Leaving the system of points, we pass to a set of innu- 
merable equal particles, distributed over the space considered. Let ¢ 
be a scalar quantity, whose values in the points A,, A,,. . . A, of 
one of the particles are q,, qs, - + + Ye, the position of these points 
and the values of g being the same in all particles, and these values 
being such that 
Gy = V2 =} st eae En Vk —= 0) . . . . . . (12) 
We proceed to determine the sum +g of the values g, belonging 
to all the points A that lie within the above mentioned closed sur- 
face 5. Of course, the particles lying completely within the surface 
will contribute nothing to this sum. Yet, it may be different from 
0, because a certain number of particles are cut in two by the sur- 
face, so that only a part of the values g,, q,, -- - ge belonging 
to each of these are to be taken into account. 
Assume in each particle an origin O (having the same position 
in each) and regard this as composed of & points O,, O,, . . . Ok. 
Attach to these the values — q,,—4q., ...—gk. Then, in virtue 
