An introduction to the mathematical theory of attraction ...

218 Electric images.
taken round the circle is - 2.r (E + e). Also at all points outside the circle, V2V = O. Hence, throughout this region
= (E + e) log, (17)
as there can be only one function satisfying the conditions
specified above.
Inside the circle the potential U of the distribution on its
circumference must satisfy the conditions V2 U= O throughout
the interior region, and
U + e log - = constant
A-L
for any point P on the circumference.
Any two functions of the coordinates satisfying these
conditions can differ only by a constant. Hence
U = e log BQ + c.
The constant c is determined from the value of V at the
circumference. From this we have
BP i1
e log + c = (E + e) log -;
whence
1 a
c = (E + e) log - -e log -,
a f
and the total potential V at any point Q in the interior of
the circle is given by the equation
= e (log Q - log + (E + e) og (18)
The formulae for a cylindrical distribution of mass in
which e and E denote charges per unit of length are obtained
from those for the corresponding uniplanar distribution by
changing e and E into 2e and 2E.
118. Total Uniplanar Mass on Curve.-If the
potential on one side of a curve be constant, and on the other
side be the same as that due to a number of charged points