An introduction to the mathematical theory of attraction ...

Graphic Representation of Lines of Force. 53
33. Field of Force Symmetrical round an Axis.When all the forces lie in planes passing through a common
axis OZ, and in any two of these planes are equal respectively, and disposed in the same manner, the whole system is
one of revolution round OZ.
In this case, points on the same line of force in their revolution trace out circles for which the inductions are equal.
Conversely if there be two points QI and Q2 lying in the
same plane through OZ, and if the induction for the circle
traced by the revolution of Q1 be equal to that for the circle
traced by Q2, then Q1 and Q2 must lie on the same line of
force. For if not, draw in the plane OQ1Q2 the equipotential
curve passing through Q2, and let it meet the line of force
through Q, in Q'2, then the induction over the portion of the
equipotential surface lying between the circles traced by Q2
and Q'2 must be zero, which is impossible except there be a
point of equilibrium between Q2 and Q'2; therefore, in general, Q2 and Q'2 are on the same line of force.
34. Graphic Representation of lLines of Force.If the field of force be such as would be produced by a finite
number of masses situated on the axis of revolution, it is
always possible to obtain any number of points for which
the value of the corresponding induction, as explained in
Art. 33, is known.
Let A be a centre of force on the axis of revolution OZ
at which the mass is m', then the lines of force for m', considered alone, are straight, and the tubes of force are cones
having A as vertex. Draw through A in a plane containing
OZ, a straight line AP2 making with AZ an angle 0'1 such
that 2rm' (1 - cos 0',) = i, and draw AP2, AP., &c., making
angles 0'6, 0',, &c., with AZ such that
1 - cos 0'1 = cos 0'î - cos 02 = cos 0' - cos 0'3 = &C.;
then since
-oI 2 a sin 'dO'dp = 2rm' (1 - cos O'),
ÇQ2 t27r
and I sin 0O'O'df = 2r (cos 0'i - cos 0'),
the induction is i for each cone of revolution whose inner