Caption: Mathematical Models by Arnold Emch - Series 1 (1920)
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This is the 6th species in the classification of ruled quartics by Cayley, and the 5th species in that of Cremona. The surface has Ci as a double line which must be counted twice. Moreover there is another double line (x = 0) in the xy — plane. J/,. Quintic, p=3, q = 2. Cartesian equation: (y* - 3x2y) z2 + (2x4 + Ax2y2 + 2y* - 2ax + Qaxy2) z + (x2 + y2 - a2)(y* - Sx2y) = 0. C\ is in a triple line o the surface and must be counted as three double-lines. The surface has the twisted cubic a(l+0 at(l-t2) a(3t+t*) x = ,y = 1—3*: l—3^2 ,z = l—3t2 as a double curve. In Schwarz's classification of quintic ruled surfaces this is marked with III A. 5. Sextic, p = 2, q = 1. Cartesian equation: (x2 + y2) (x2 — y2)2 - [2xyz — a(x2 - y2) ] 2 = 0. C\ is a quadruple line of the surface and must be counted as eight double lines. There are two double generatrices x + y = 0 in the xy—plane. Tangent Surfaces of a Rectilinear (292)-Congruence The joins of corresponding points z' and z of two complex planes Ey and E in space, related by ax + b z' = , ab — bc == 0. ! cz+d form a congruence. The elements of such a congruence touch a certain quartic surfaces Q. Plaster cast models of two such surfaces have been constructed: 6. Q[ with a system of elliptic sections. (C. N. Stokes) Cartesian equation: (x2 + y2 + z2) (x2 + % 2 ) — 36 (x2 + y2) = 0
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