. Differential and integral calculus, an introductory course for colleges and engineering schools. n surface by planes parallelto the coordinate planes. Consider first the plane x = k. Wesubstitute k for x in the given equation, and get, after reduction, (a) y b2 l -. + <-5) 1. This may be regarded as the equation of the right cylinder whichprojects upon the yz-plane the curve of section S, made withthe surface by the plane x = k, or, as the equation of the projec-tion, Sf, of this curve of section upon the 2/2-plane, or, as the equa-tion of S itself. This curve of section S is, therefore,
. Differential and integral calculus, an introductory course for colleges and engineering schools. n surface by planes parallelto the coordinate planes. Consider first the plane x = k. Wesubstitute k for x in the given equation, and get, after reduction, (a) y b2 l -. + <-5) 1. This may be regarded as the equation of the right cylinder whichprojects upon the yz-plane the curve of section S, made withthe surface by the plane x = k, or, as the equation of the projec-tion, Sf, of this curve of section upon the 2/2-plane, or, as the equa-tion of S itself. This curve of section S is, therefore, an ellipse * I a I is read absolute value of a, and denotes the numerical magnitude of awithout regard to sign. For example, | — 5 | =5, and I +5 | =5. The expres-sion J z I := I c ] is therefore equivalent to the longer expression —c=z%-\-c. §185 THE POINT, THE PLANE, AND THE SURFACE 273 whose center is in the z-axis, whose axes lie in the zz-plane and / k2the z?/-plane, and whose semi-axes have the lengths b\/l -^ When | k | \ a \ 04:. it is imaginary. Therefore every section of the ellipsoid by aplane parallel to the 2/0-planeis an ellipse, real or imagi-nary. As k increases from 0to a, or decreases from 0 to— a, the cutting plane moves —away from the origin, and theellipse S grows smaller andsmaller, until, when a;=±c,it has shrunk to a point of thez-axis. The solid inclosed by the surface may therefore beregarded as made up of a series of elliptic slices strung on thex-axis. Exactly similar arguments show that every section ofthe ellipsoid by a plane parallel to the zz-plane, or parallel to thexy-pl&ne, is an ellipse, real or imaginary, and that the solid maybe regarded as made up of a series of elliptic slices strung onthe i/-axis, or as elliptic slices strung on the 2-axis. We have nowa good idea of the shape of the surface. It can be proved thatevery plane cuts the surface in an ellipse, but we shall not givethe proof. The surface (and the solid bounded by the
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1912
