An introduction to the mathematical theory of attraction ...

118 The Potential.
principle of the Conservation of Energy, Pds == Pds', and also, Qds = Q'ds',
P p'
whence, in going from one line of force to another, - = -. Hence, throughout
P
6, the ratio - is constant.
6. If two distributions, A and B, of mass have throughout a region 5 to
which they are both external the same equipotential surfaces, the resultant
forces due to these distributions are throughout the region @ codirectional and
in a constant ratio.
Throughout 6, if l and V be the potentials due to A and B, we have
U=f=(V) and also v2U= 0, 2V = 0. Hence U= V t c where c and c are
constants.
7. If two distributions, A and B, of uniplanar mass, acting with a force
varying inversely as the distance, have the same closed curve s surrounding both
as an equipotential, every curve outside s which is an equipotential for A is also
an equipotential for B.
Let Mi be the total mass, and V1 the potential corresponding to A, and M2
and T2 those corresponding to B, and let the values of Vl and V2 at s be Ci and
1M2
C2; then, putting- - = K, we have iKM\ + M2 = 0.
sMi
Now imagine a distribution A' whose elements occupy the same positions as
those of A, but such that any mass m' belonging to it is given by the equation
m' = Km, when m is the corresponding mass belonging to A.
Let the distribution A' coexist with B, and let V be the corresponding
potential; then at s we have V = K Ci + C2; and since V is constant and the total
mass zero, by Art 36, we get
dFds = 0,
r v =o,
dv
when the integral is taken round the curve s. But by Art 44, the integral
dVd
dv
taken round a circle at infinity is zero; therefore, by (13), Art. 60, V is constant throughout the whole plane outside s, andbeing zero at infinity is therefore
zero. Hence since KF + V2 = V = 0, the distributions A and B have the
same equipotential surfaces everywhere in the plane outside s.
65. Potential a Maximunm or a iiinimum.-If the
potential V of any system of mass be a maximum at a point
P, there is positive mass at P, and if the potential be a
minimum, there is negative mass.
To prove this, describe a small sphere S round P as
centre, and take P for origin; then if V be a maximum at
dVis negative,
P, at each point of the surface of this sphere dr is negative,
dr