84 M. M. SLOTNICK 
VARIATION OF CURVATURE WITH A 
Equation (11) expresses the algebraic value of the curvature at 
the point P of that plane normal section C of the surface S, made by 
the plane through the z axis whose angle with the (x, z) plane is X. 
Varying the value of \ from o° to 180° continuously corresponds to 
twisting the normal plane continuously around the z axis from the 
(x, z) plane, (A=o°), through its position of coincidence with the 
(y, z) plane, (A=go°), and finally back to the (x, z) plane for which 
A=0° or 180° (from 180° to 360°, the sections obtained as } varies 
from 0° to 180° are repeated). 
The first fact to notice is that if for all values of \ from o° to 180°, 
the curvature (11) is always positive, or always negative, the surface 
in the neighborhood of P lies on one side of its tangent plane. If, 
however, as A varies in this range, the curvature changes sign, part of 
the surface lies on one side of the tangent plane to S at P and part on 
the other. The latter situation is the one shown schematically in 
Figure 1. 
Suppose that we fix our attention to the curvature k; at P of the 
section for which A=\,; that is, 
(12) ky = rcos?)\; + 2s sin \; cos Ay + ¢ sin? Ay. 
Consider the curvature k2 at the same point, of the section whose 
plane is at right angles to that of the first section; that is, that for 
which A=A2=Ai + 90°. Substituting this value for \ in (11) yields: 
(13) k. = rsin?X\ — 2ssin\ cosA + #£ cos? \X. 
The sum of (12) and (13) is: 
(14) kitk=r+t, 
a constant. To put this interesting result in words: The algebraic sum 
of the curvatures at a non-singular point of an analytic surface of any 
two normal plane sections at right angles to each other is constant. One- 
half of this constant is called the mean curvature of the surface at the 
point. 
The next question to be raised in regard to (11) is whether, as A 
varies from o° to 180°, the curvature k reaches extreme (maximum 
and minimum) values; and, if so, for what values of \? A maximum 
or minimum value of & is obtained whenever — =o and De” O. 
Differentiating (11), we obtain: 
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