CURVATURE OF EQUIPOTENTIAL SURFACES 83 
and 
d*z 
(9) on = (r cos?\ + 2ssind\ cosA + ésin?A) + - 
The terms of each of these series beyond those written involve R to 
at least the first power. At P, then, where z= R=o, 
(10) e oe *r>+ in + #sin?r 
10 — =bd, — =rcos 25 sin \ cos sin? d. 
dR) dR 
It is these values, as we have seen, that we must use in (7) to ob-. 
tain the curvature of C at P, and substituting these, we have 
(11) k =rcos?\’ + 2ssin\cos\ + ésin?v. 
In ordinary work in the elementary differential calculus, the value 
of the curvature of a curve at a point is taken as the absolute value 
of (6) and no attention is paid to the fact that the numerator, 
qa? May be negative as well as positive. However, in surface theory, 
the algebraic value of the curvature is important, and our definition 
of the curvature of the plane section C of the surface S at the point 
P, as embodied in (7), places due regard on the algebraic value of 
7 =e We append a schematic series of figures here to show, somewhat 
more Clearly, what is meant. 
z 

2a 2¢ 
Fic. 2 
Figure za indicates the form of the curve C in the neighborhood 
of P, if, for the value of d defining C, = >o. Figure 2b indicates the 
same thing if 
dz d°z 
dR? =o and— 7 UR” ° at P, the form of C is as shown in one or the other 
of the figures 2c. 
me Finally, if for that value of » defining C, 
413 
