CURVATURE OF EQUIPOTENTIAL SURFACES! 
M. M. SLOTNICK? 
Houston, Texas 
ABSTRACT 
If one is to have a complete and satisfying understanding of the theory of the tor- 
sion balance, it is essential to have clearly in mind a physical picture of the quantities 
involved. Most authors assume either that their readers are well versed in the differen- 
tial geometry of surfaces in a Euclidean three-dimensional space or that a discussion of 
surface curvature is far beyond the reader’s ability to grasp. Too often is neither of 
these true. The writer attempts to present a mathematical discussion of this matter in 
which only the bare fundamental concepts of the differential calculus are needed. 
INTRODUCTION 
In this paper, the writer attempts to develop the mathematical 
theory of the curvature of the equipotential surfaces due to the 
gravitational field, with the purpose of producing a clear “‘physical” 
picture of this quantity. It is the writer’s impression that such a de- 
velopment, although fundamental to the mathematician, is lacking 
in the usual torsion balance literature available to the geophysicist. 
Our goal is to find the curvature relations used in torsion balance 
work. The mathematical needs for this subject are not too great— 
and for further simplification, the demands of mathematical rigor are, 
at times, sacrificed in this paper 
DEFINITIONS 
Consider an arbitrary point P on a surface S, which is “‘smooth” 
in the neighborhood of P in the sense that the equation of S may be 
expressed as a series when the surrounding space is referred to a 
convenient cartesian codrdinate system. The tangent plane to S at 
P is that plane in which all the lines tangent to S at P lie. The straight 
line perpendicular to this plane at P is the normal to the surface S at 
that point. 
COORDINATE SYSTEM 
We now proceed to choose, as is our privilege, a codrdinate system 
in such a manner that the mathematical work involved is materially 
' Read before the Association at the Oklahoma City meeting, March 25, 1932. 
* Geophysics department, Humble Oil and Refining Company. Introduced by 
L. W. Blau. 
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