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 
Fic. 1 
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 codrdinate 
system, is 
(1) z= 2(x, y)s 
and that this expression is expanded in series about P; that is, 
@) 8 Cae ees (BN eG are SY ae I) am 
where 

02 Oz 
PS an diz ai 
022 02z 072 
in ax? fla axdy ei ay? 
the values of these derivatives being taken at the point P, whose 
codrdinates are, of course, (0, 0, 0). 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:(0, 0, o), equation (2) must 
be satisfied by its codrdinates. Accordingly, a=o. Again, the slope 
of the curve on S at P, which is the section of the surface made by 
411 
