An introduction to the mathematical theory of attraction ...

Surface Distribution. 125
Again, if P be a point in the region @i, and r denote the
distance of any point from P, by Art. 59, we have
1rdi dS d() dS - 47r=p.
jr dv, J ' dvi\r
Also, by Art. 58, we have
- 1 ddS = 1 - - dS.
J r dv2 J dv2 r
Adding this equation to the former, and remembering
that at the surface S we have b= = =, and that
d d
- = -, we get
dvi dV2
[(dL + dv ldS = - 47Trp. (15)
r dvi dv2) 4 (15
Hence at any point in ~i the function p expresses the
potential of a surface distribution whose density is
1 (dl) dc.p
47r \dv dv2J
In like manner ~ is the potential in ~2 of the same surface
distribution. Also, if M be situated in (2 we have f -V
throughout 1i, and if M be situated in 5, we have t = V
throughout ~2. IHence the surface distribution is determined
which produces the same potential on one side of S as a
given mass distribution existing on the other side.
When the mass M is inside S it is equal to the total mass
of the surface distribution.
It is obvious that we can show in a similar manner, that
if space be divided by a boundary or set of boundaries into
two regions, it is always possible to distribute mass over the
boundary so as to produce in one region the same potential
as that produced by a given distribution of mass existing in
the other region. The density of the required surface distribution is determined in the same manner as before.
Another theorem, in some respects more general than
those given above, is the following:It is always possible to distribute mass over a surface S
closed or open so as to produce a potential which is equal at
each point of S to a given function of the coordinates.