82 M. M. SLOTNICK 
the (x, z) plane, is the value of p there, and consequently this is zero. 
Similarly, g=o. Thus, by virtue of our choice of axes, the series (2) 
reduces to 
(3) 2 = W(rx? + asxyt+ty?)+---. 
CURVATURE OF A PLANE SECTION OF S AT P 
Let C be the curve of.intersection of any plane zPR through the 
z axis with the surface S, and let the angle between this plane and the 
(x, z) plane be A. Obviously, then, the curve C is a plane curve and 
PR, the tangent line to it at P, lies in the (x, y) plane, with the angle 
xPR equal tod. The lines Pz and PR are at right angles to each other, 
and we choose these lines as céordinate axes in their plane zPR. Since 
(4) x = RcosiX, y = Rsini, 
the equation of C in terms of the codrdinates (R, z) is, by virtue of (3): 
(5) z = YR*rcos?\ + 2ssindcos\ + ¢sin?A)+---. 
We recall at this stage the theorem of the elementary differential 
calculus that the curvature of the curve whose equation is y=f(x) 
at a point is the value of 
d*y 
dx? 
PHS) | 
at that point. In equation (5), z is expressed as a function of R, and, 
at P, z=R=o. The quantities 7, s, are constants, inasmuch as they 
represent the values of the second derivatives of z with respect to x 
and y at P; and 2 is a constant for the particular section C chosen. 
Consequently, the curvature of C at P is the value of 
d*z 
dR? 
dz 2 3/2 : 
i+(Ge) | 
dR 
when z= R=o. 
Differentiating (5), we obtain 
dz 
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(6) Be 
(7) k= 
412 
