ACTING INVERSELY AS THE SQUARE OF THE DISTANCE. 16] 
passage through the surface of the sphere suddenly undergo the 
alterations s fe a —— =, &e. 
aX aX aZ eV FV ENV 
dx dy tet ae dy? + 2 
within the sphere = — 474, and in external space = 0. ‘But 
on the surface itself it loses its simple signification: speaking 
precisely, we can only say that it is an aggregate of three parts, 
each of which has two different values ; and thus there are, pro- 
perly eight combinations, one of which agrees with the values 
within the sphere, and another with those without the sphere, 
whilst the remaining six have no signification whatsoever. The 
analysis by which some geometers have derived for the sur- 
face of the sphere the value — 27k, or the mean between the 
internal and external values, cannot, I think, be regarded as sa- 
tisfactory, if the idea of differential pieiietenta is conceived in 
its mathematical purity. 
becomes 
9. 
The result obtained in the preceding example is only a par-_ 
ticular case of the general theorem, according to which, if the 
point O be situated in the interior of the acting mass, the value 
2 2 
of » + uae - a is equal to the product of — 4 7 multi- 
plied into the density at O. The most satisfactory basis of this 
important theorem appears to be the following. 
We assume that within ¢ the density k nowhere varies dis- 
continuously, or that it is a function of a, 6, c represented by 
JF (4, 6, c), the value of which varies continuously everywhere 
within ¢, but without ¢ becomes = 0. 
Let ?' be the space into which ¢ is changed if the first coordinate 
of each point of the bounding surface be diminished by the 
quantity e; or, which is the same thing, if the bounding sur- 
face be moved by that quantity towards the origin of co-ordinates 
in a direction parallel to the first coordinate axis; let ¢ consist 
of the spaces ¢° and 4, ¢! of ¢° and §', so that 2° shall be the whole 
space common to ¢ and 7. 
We have to consider the three integrals. 
fe J (a, b,c) (a — x) dt ; 
(a@—a)? + (b—y2?+(e—z3 °° * (1.) 
a, b,c) (a—x# —e)dt 
Se Se - + (2) 
