Early geophysical papers of the Early geophysical papers of the Society of Exploration Geophysicists earlygeophysical00soci Year: 1947 CURVATURE OF EQUIPOTENTIAL SURFACES 81 simplified. Let the normal to S at P, oriented in either of its direc- tions, be the z axis. In the tangent plane to S at P let any two mutu- ally perpendicular lines be chosen as the x and y axes. Both of these axes are obviously perpendicular to the z axis. Let us also point out Fig. i that the tangent line at P to any curve on S passing through P lies in the (x, y) plane. The situation thus far described is schemati
Early geophysical papers of the Early geophysical papers of the Society of Exploration Geophysicists earlygeophysical00soci Year: 1947 CURVATURE OF EQUIPOTENTIAL SURFACES 81 simplified. Let the normal to S at P, oriented in either of its direc- tions, be the z axis. In the tangent plane to S at P let any two mutu- ally perpendicular lines be chosen as the x and y axes. Both of these axes are obviously perpendicular to the z axis. Let us also point out Fig. i that the tangent line at P to any curve on S passing through P lies in the (x, y) plane. The situation thus far described is schematically shown in Figure 1, where, for convenience, only part of the surface S in the neighborhood of P is indicated. EQUATION OF S Suppose that the equation of S, referred to the chosen coordinate system, is (i) 2 = z(x, y), and that this expression is expanded in series about P; that is, (2) z = a + (px + qy) -f- ^(rx2 + 2sxy + ty2) + ⢠⢠â , where dz dz ox dz dy dx* d2z dxdy t dy* the values of these derivatives being taken at the point P, whose coordinates are, of course, (o, o, o). The series (2) has been written only to the second order terms, which are all that are required. Since the surface S passes through P:(o, o, o), equation (2) must be satisfied by its coordinates. Accordingly, a = o. Again, the slope of the curve on 5 at P, which is the section of the surface made by 411
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