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By Andryan A. A.

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The relation of the constants A, B to the semiaxes a, b of the conic is For details the reader is referred to [31]. In addition to representation of specific curves, it is important to note that natural equations allow us to NATURAL EQUATIONS AND INTRINSIC GEOMETRY 35 pass fairly easily from a curve to some of its geometrical transforms: parallel curves, evolutes, pedals, roulettes and so on [31]; in this connection the reader may recall the contents of vol. 1, pp. 35-38. Further, natural equations might be considered in connection with pursuit.

From these we determine c and r in terms of R = 1/ K and t = 1/T: c=x+Rn+TR 1 b, r 2 =R 2 +T2 (R 1) 2 • This determines completely the osculating sphere at a point of C. Now, a necessary and sufficient condition for C to lie on a sphere is that r 2 = R 2 + T 2 (R 1} 2 = const. for we see then, by differentiating, that C1 = 0 so that C has at all points the same radius and center of the osculating sphere, hence it lies on a sphere. By using the Taylor series together with the Frenet-Serret formulas we can write down the local power-series for the Cartesian coordinate in the neighborhood of the point corresponding to s = so; this enables us to study the local shape of the curve: 2 I K KK 4 3 x=(s-so)--(s-so) --(s-so) +· · · 6 2 8 K K 3 KII -K 3 -KT 2 4 2 y=-(s-so) +-(s-so) + (s-so) +· · · 2 6 24 I KT 3 2K 1 T+ KT 1 4 z=--(s-so) (s-so) +··· 6 24 (here x is used as a coordinate whereas it was used before as the radius vector).

Thus we have a simple mechanical arrangement for drawing compound roulettes of an ellipse (see vol. 1, p. 37) with respect to a plane curve C. When C is a straight line we get the simple roulette (=elliptical equivalent of the cycloid); this was used by R. C. Yates to generate a minimal surface of revolution [186]. Similar device of a vertical wheel is also used in the Amsler planimeter, an analog linkage device for measuring plane areas, which we shall now describe. Let a rigid straight bar PtP2 of length l move in the plane so that the end points are Pt(Xt, Yt) and p2(x2, y2), and the bar makes angle 8 with a LINKAGES AND OTHER ANALOG MECHANISMS 19 F3 (a) (b) (c) Fig.

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A Boundary Value Problem in a Strip for Partial Differential Equations in Classes of Tempered Functions by Andryan A. A.


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