An introduction to the mathematical theory of attraction ...

54 Lines of Force.
and outer boundaries are generated by successive lines drawn
as above. Hence, if '1, I'2, &c. be the inductions due to m'
for circles traced by the revolution of points on AP,, AP2,
&c., we have
I' = i, I' = 2i,.... I',, = n'i.
In like manner for a mass m" situated at a point B on
the axis we have
Ir =, 1"2 2i,...., =n"i
where I"', &c. correspond to points on lines BQ1, &c. drawn
through B such that
27rm" (1 - cos 0") = i, 1 - cos 0"1 = cos 0" - cos 0"2 = &c.
If now the field of force be due to the joint action of m'
and m", and we take the point R in which the straight line
AP,n intersects the straight line BQn", for the induction I
corresponding to R, we have
I = In' + IT"n" = (n' + n") i;
and if the numbers n' and n" vary, but so that n' + n" = constant = n, we obtain a set of points for which the corresponding induction is ni. A curve passing through these points is
a line of force for the joint action of the two masses m' and
n". The points of intersection of these lines of force with
the straight lines bounding the tubes of equal induction for
a third mass, determine in a similar manner points for which
the corresponding inductions are equal in the case of the joint
action of three masses. Thus a line of force can be drawn
for this case, and it is plain that the method can be extended
to the case of any number of masses situated on the axis of
revolution.
This mode of obtaining a graphic representation of the
lines of force is given by Clerk Maxwell, "Electricity and
Magnetism," Part I., Chapter vii.
35. Force acting on Elenent of Surface of
charged Conductor.-When a conductor is charged with
electricity in equilibrium, the electricity in each element of
its surface is, in general, acted on by a force normal to the