CURVATURE OF EQUIPOTENTIAL SURFACES 87 
Let us note that the difference of equations (12) and (13) may be 
written 
(19) k, — ko = (r — #) cos 2A + 25 sin 2X, 
and, using the results of (17), we find that the extreme curvatures 
satisfy the relation 
(20) eee eee ,[T 7 |- a apAt 
Ww 
APPLICATION TO EQUIPOTENTIAL SURFACES 
The torsion balance deals with the equipotential surfaces, which 
are the surfaces perpendicular to the lines of force of the earth’s 
gravitational field. Let the equation of the equipotential surface 
through the center of gravity of the suspended system of a torsion 
balance be 
(21) U(x, y, z) = constant, 
with respect to the usual coérdinate system used; namely, the z axis 
is the direction of the force of gravity at the center of gravity of the 
suspended system (taken as the origin) directed positively downward, 
the x and y axes are the lines through the origin, tangent to the equi- 
potential surface, bearing north and east respectively. The value of 
gravity at the point is, then, 
(22) RSs 
The torsion balance observes the values of 
aU dg aU ag 





2. = — = 
(23) 0x02 ax. Oyoz Oy 
and 
Ge) 5 0?U 07U a 07U 
24 = + z cd 
Oy? Ox? ; ; axdy 
* It will be noticed that the mathematical analysis does not differentiate between 
k, and kz, as to which is the maximum and which the minimum, and cannot do so. This 
becomes obvious when we consider that if, initially, the z axis had been oriented in the 
opposite direction, as it well might have, the maximum curvature would become the 
minimum and vice versa. This corresponds with a reflection of the curve of Figure 3 
in the \-axis; for what was originally z is now —z and the equation (10) changes sign. 
In short, equation (20) states that the numerical value of the difference between the 
principal curvature is w. In the numerical work used in torsion balance calculations, 
however, the curvatures assume, tacitly, numerical algebraic values and the cor- 
responding ) to each of these can be assigned. This will be indicated briefly in the last 
paragraph of the paper. 
417 
