An introduction to the mathematical theory of attraction ...

122 The Potential.
of the degree m + 1, &c.; and, as V2 rV= O for all values of the
coordinates, v2H, = 0, and therefore equating to zero the
coefficient of the highest power of z in V2-Hm, we get
/d2 d2\
\dX2 + CYIZJ 0.
+ dy2) un = ~.
If p be the perpendicular from any point on the axis of z, we
have un = p=f(p); and by equation (18), Art. 48, we get
d2 d 1 _ 2 d2 _2 )
dæ dy2? dIp dp2+
whence we obtain
d + nf = 0;
and therefore
f/() = A sin (n + ~).
The value of p for a tangent plane to the equipotential
surface is given by the equation f () = 0, and therefore
no + a = s7r when s is an integer.
a wr a 7w
Hence i = -, = - - - &c., and 2- 1 = -that
is, two adjacent sheets of the equipotential surface intersect
at the angle -.
In the case of uniplanar mass acting inversely as the
distance, it can be shown in a similar manner that if an equipotential curve in a part of the plane unoccupied by mass
have a multiple point of the order n, the angle between two
adjacent tangents at the point is r.
69. Biagraems of Equipotential Surfaces.-If the
value of the potential at a point P be known for each of
the portions of mass of which a system is composed, we can
find the potential of the whole system by addition. This
principle enables us to construct the equipotential curves due
to a set of centres of force of given intensity. If the centres
of force be situated on the same straight line L the equi