An introduction to the mathematical theory of attraction ...

Historical. 65
CHAPTER IV.
THE POTENTIAL.
SECTION I.-Elementary Properties.
39. Historical.-When a particle, whose mass is unity, is
moving under the action of a force whose components X, Y, Z
at any point are functions of its coordinates, the velocity v of
the particle is given by the equation
v = 2 (Xdx + Ydy + Zd) + C.
If the quantity under the integral sign is a perfect differential we obtain, by integration, a function U of the coordinates
which is called the force function.
Laplace seems to have been the first to employ the
force function in the solution of questions relating to attractions. The powerful analysis of Laplace led to many results
of great value, and showed the importance of a study of the
properties of this function. The research was taken up
with splendid success by Green, whose great theorem may be
said to dominate the whole field of the higher Mathematical
Physics, and to whom the term Potential is due.
Lagrange, in the "Mécanique Analytique," made frequent
use of the force function which corresponds to a material
system and which is obtained by integration from that
belonging to a particle; but in his Equations of Motion
in generalized coordinates he substituted another function
which is equal to the force function with its sign changed.
This latter function has an important physical meaning, as it
expresses the potential energy of the moving system. To it,
therefore, the term potential can be applied more properly
than to the force function.
40. Definition of the Potential. —The potential of a
mass system at any point is the energy due to the mutual
action of the unit of mass placed at the point, and the system,
regarded as invariably connected, placed in its actual position.
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