THE EtiTVQS TORSION BALANCE

83

these two planes are mutually perpendicular.1 We shall refer
to these as the "principal planes." For the moment let us
consider the special case in which the x- and y- axes are taken in
the directions of these principal planes. We can then represent
the curved level surface and the principal planes as in Fig. 45.
Let the direction of the vertical and the value of gravity at the
points a, &, c, d, be ga, g^ gc, gd, respectively. Also, let pi and />2

X(NORTH)

FIG. 45.—Curved oquipotential surface and horizontal gravitational forces.

be the radii of curvature in the two principal planes. Now,
remembering that the gravity vector is always perpendicular to
the level surface, and considering the horizontal dimensions of the
figure as small differentials over which the second derivatives of
the potential are constant, we have the following relations:

The small horizontal gravity component gv is the rate of change

times

[ d /dU\ 1
of the horizontal force in the ^/-direction — ( — } = Uyv
1 For a proof using only the fundamental ideas of the differential calculus,
see Slotnick. 1932.