This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1893 edition. Excerpt: ... 32). Considering a line defined by its "six coordinates" (a, b, c, f, g, h), the condition which expresses that the line shall belong to an involution is (A, B, C, F, G, Hla, b, c, f, g, h) = Q, where (A, B, C, F, G, H) are arbitrary coefficients; if they are the coordinates of a line, that is, if AF+ ...
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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1893 edition. Excerpt: ... 32). Considering a line defined by its "six coordinates" (a, b, c, f, g, h), the condition which expresses that the line shall belong to an involution is (A, B, C, F, G, Hla, b, c, f, g, h) = Q, where (A, B, C, F, G, H) are arbitrary coefficients; if they are the coordinates of a line, that is, if AF+ BG + CH = 0, then the condition expresses that the line (a, b, c, f, g, h), instead of belonging to a proper involution, meets the line (F, G, H, A, B, 0). 1 The theory is explained in ray memoir "On the Six Coordinates of a Line," Camb. Phil. Trans, vol. xi. 1868, 348. In explanation of the subsequent analytical investigations of the present memoir, it is convenient to remark that if on a given line we have the two points (a, /S, y, t) and (a', /J1, y', &), and through the given line two planes Ax + By + Cz + Dw = 0 and A'x + B'y + Cs + D'u = 0; then we have /S7'-fly:7a'-y'a: a/9'-o'/3: aV-a'S: /3'-fli: yV-y'i = Ajy-A'D: Bjy-B'D: CD'-CD: BC-B'C: CA'-CA: AB'-A'B; and denoting either of these sets of equal ratios by a: b: e: /: g: h, then (a, 6, c, /, g, h) satisfy identically the relation a/+bg + ch = 0, and are said to be the six coordinates of the line. then the coordinates (a, b, c, f, g, h) of the reciprocal curve will satisfy the condition of involution (F, G, H, A, B, Ga, b, c, f, g, h) = 0. The reciprocal of the before-mentioned skew cubic x: y: z: w =1: t: t2: & is the quartic torse having for its edge of regression the skew cubic 3XZ--Y3 = 0, YZ--9XTP = 0, 3YW-Z2 = 0; or, what is the same thing, the skew cubic X: Y: Z: W = 1: 3t: 3P: &; see my paper "On the Reciprocation of a Quartic Developable," Quart. Math. Journ. vol. vii. (1866), pp. 87--92, 372. 68. Hence the reciprocal of the...
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