607] 
A MEMOIR ON PREPOTENTIALS. 
335 
two surfaces, so as to give rise to a series of consecutive material surfaces; the 
quantity of such matter is infinitesimal, and the density of each of the material surfaces 
is therefore also infinitesimal. As the attracted point comes from the external space 
to pass through the first of the material surfaces—suppose, to fix the ideas, it moves 
continuously along a curve the arc of which measured from a fixed point is = s—there 
is in the value of V (or, as the case may be, in the values of its derived functions 
like as the attracted point passes through the different material surfaces respectively. 
Take the case of a potential, q = — ^; then, if the surface-density were finite, there 
would be no finite change in the value of V, but there would be a finite change 
in the value of -j—; as it is, the changes are to be multiplied by the infinitesimal 
density, say p, of the material surface; there is consequently no finite change in the 
value of the first derived function; but there is, or may be, a finite change in the 
d 2 V . . 
value of -j- and the higher derived functions. But there is in V an infinitesimal 
change corresponding to the passage through the successive material surfaces respectively; 
that is, as the attracted point enters into the material space, there is a change in 
the law of V considered as a function of the coordinates (a, .., c, e) of the attracted 
point; but by what precedes this change of law takes place without any abrupt 
change of value either of V or of its first derived function; which derived function 
may be considered as representing the derived function in regard to any one of the 
coordinates a, .., c, e. The suppositions, that the density outside the bounding surface 
was zero and inside it constant, were made for simplicity only, and were not essential; 
it is enough if the density, changing abruptly at the bounding surface, varies con 
tinuously in the material space within the bounding surface*. The conclusion is that 
V', V" being the values at points within and without the bounding surface, V' and 
V" are in general different functions of the coordinates (a,.., c, e) of the attracted 
point; but that at the surface we have not only V' = V", but that the first derived 
functions are also equal, viz. that we have 
27. In the general case of a Potential, we have 
If p does not vanish at the attracted point (a,.., c, e), but has there a value p' 
different from zero, we may consider the attracting (s + l)-dimensional mass as made 
* It is, indeed, enough if the density varies continuously within the bounding surface in the neighbourhood 
of the point of passage through the surface; but the condition may without loss of generality be stated as 
in the text, it being understood that for each abrupt change of density within the bounding surface we must 
consider the attracted point as passing through a new bounding surface, and have regard to the resulting 
discontinuity.