A Treatise on the Analytic Geometry of Three Dimensions Volume 1
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1882 edition. Excerpt: ... the point comes twice into the same position. Reciprocally, the system may have O double planes; viz., considering the developable as the envelope of a plane, if in the course of its motion the plane comes twice into the same position, we have a double plane. These singularities will be taken into account if, tin the formnlaa of Art. 326, we write r=g+ G instead of r = g, and in the formulae of Art. 327, write B = h + M. In like manner, the system may have v stationary lines, or lines containing three consecutive points of the system. Such a line meets in a cusp the section of the developable by any plane, and accordingly, in Art. 326, instead of having K = m, we have = m + v; and, in like manner, in Art. 327, instead of t = n, we have i = n + v. Once more, the system may have to double lines, or lines containing each two pairs of consecutive points of the system. Taking these into account we have, in Art. 326, 8 = x + a, and in Art 327, r = y + w. 329. To illustrate this theory, let us take the developable which is the envelope of the plane of + kbtf1 + $k (k-1) ctT + &c. = 0, where t is a variable parameter, a, b, c, &c. represent planes, and k is any integer. The class of this system is obviously Jc, and the equation of the developable being the discriminant of the preceding equation, its degree is 2 (k--1); hence r = 2 (k--1). Also it is easy to see that this developable can have no stationary planes. For, in general, if we compare coefficients in the equations of two planes, three conditions must be satisfied in order that the two planes may be identical. If then we attempt to determine t so that any plane may be identical with the consecutive one, we find that we have three conditions to satisfy, and only one constant / at our...
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