An introduction to the mathematical theory of attraction ...

66 The Potential.
The potential of a system of repelling or attracting
masses at any point may, perhaps, be more simply defined as
the work done against the forces of the field, due to the system,
in bringing the unit ofmass to the point from an infinite distance,
the positions of the acting masses being supposed invariable.
Since no energy is expended in bringing an invariable
system unacted on by externalforce into any assigned position,
it is plain that one of these definitions is equivalent to the other.
When we have to do with attractive masses the potential,
as defined above, is negative. Hence, in the case of a gravitation potential, if the acting masses be regarded as positive,
it is simpler to define the potential, at any point, as the work
done by the forces of the system in bringing the unit of mass
from an infinite distance to the point. The potential is then
the same as the force function, and is the function which was
employed by Laplace. In questions relating to gravitation
the potential is still commonly so regarded.
It would seem to be the simplest method in all cases to
adopt as the definition of the potential the second of those
given above; and if the acting mass be attractive to regard it
as negative, the unit mass acted on being always positive.
In this way the algebraical expression for the potential will be
the same as that for the function used by Laplace; and when
we have to consider only the potential and the resultant force,
the same algebraical formulae will be equally valid for electric
and gravitational masses. It must, however, be remembered
that there is no mathematical artifice by which one algebraical
formula, interpreted in the same manner, can be made to
express the two physical facts, that masses of like kind repel
one another in the case of electrical action, and that they
attract one another in the case of gravitation.
When the distribution of mass is cylindrical, the potential,
as defined above, is infinite, and this is true also for- the
corresponding uniplanar distribution.
For such a distribution, if X and Ybe the components of
the resultant force at any point of the uniplanar field, the potential V may be defined by the equation V = - f (X dx + Ydy),
where no constant is to be added. This definition is provisional
only, and is to be regarded as relative to the mode of arriving
at the form of the potential given in the next Article.