An introduction to the mathematical theory of attraction ...

Examples. 229
Invert from 0, then
/,\4 K
' b ) R4
Hence the circle c', which is the inverse of c, being of uniform density, acts at
external points as if its mass were concentrated at its centre A', and the potential
U' due to c' together with - m at A' is zero outside c', and therefore, at 0, also
the total mass producing U' is zero. Hence, by (25), if P be any point
outside c, we have UP = 0, and therefore, the potential in external space due to
c is the same as that due to a mass mn placed at A, the image of O in c.
2. Uniplanar mass mn is distributed in the region outside a circle c, the areal
density varying inversely as the fourth power of the distance from a point O
inside c. Show that ms acts inside c as if it were concentrated at A which is
the image of O in c.
Invert from O, and we obtain a circle c' of uniform density. If U' be the
potential due to c' together with a mass - placed at its centre A', the
potential U' is zero at any point P' outside c'. Hence, since the total mass producing U' is zero, by (25), we have U constant throughout the region inside c,
and therefore, n at A produces the same effect inside c as the original distribution outside c.
3. Uniplanar mass is distributed over the boundary of the larger segments of
two circles cutting at an angle of 60~ so as to produce a constant potential L
throughout the interior region; find the potential at any external point, and the
distribution of mass.
Let the spherical sections in the figure of Ex. 5, Art. 119 represent the circles;
then if a charge 17 be placed at the points Il, Ji, and I3, and a charge - 7u at I2
and J2, the potential produced is constant at the circular boundary, and if it be
equal to L the charges at I1, &c. produce in external space the same potential as
the actual distribution. Ience v is determined by the equation
a a a + b
L = jlog -- - log --- a = log a- +b
I1\J\ Il J2; C2
If V be the potential at any external point Q, we have
V= {log I2Q + log J2Q - log IiQ - log J'Q - log I3Q}
I2Q. 2Q
= 1log IQ J Q 3Q
The density v of the distribution at any point P on the circle whose radius
is a is given by the equation
2rv = - c2 a2 - I1J22 }
and the total charge Ea on the arc of this circle by the equation
27rEa = 7 {a2 + a3 + J1i - al - B2},
where ai, a2 as, /3, 32, are the angles which the chord of intersection of the
circles subtends at the points I1, I2, 13, J1, J2, respectively.