Repository: UIHistories Project: Mathematical Models by Arnold Emch - Series 2 (1923) [PAGE 7]

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Let the vertices of two right circular cones lie in the #jy-plane at (0, 0, 0) and (a, 0, 0) and let the plane z = 1 cut these cones in two circles with radii r\ and r i respectively, so that the parametric repre< sentation of the first circle is l-/2 2/ * = r l w , . y = r1T-F7r Let the points of the second circle be projectively related to those of thefirst,in such a manner that the directions of corresponding radii are at right angles to each other, with angles 0 and 6 + f , respectively. T h e parametric equations of the second circle are therefore It l-/2

The tangent planes to the two cones at projectively corresponding points (generatrices) are easily established as (x +riz) t2 — 2yt—x+riz = 0, {y+rtf) t2+2 (x — a) t — y + r 2 z = 0 . For every value of the parameter / the two planes intersect in a generatrix of a ruled surface whose equation is obtained by the elimination of/, and which m a y be transformed to the form: M+ri)(x2+y2)]z2-(x2+y2+ax)2=0. T h e circle in the #y-plane through the vertices of the two cones is a nodal curve of the quartic. T h e %-axis is a nodal line. 21, Ruled Quartic. Similar to N o . 20, except that the angles of corresponding radii are 6 and3-f- 6, Fig. 3. T h e equation of this quartic is [(rix+rzy — an)2+ (r2x+riy)2]z2— (x2—y2—ax)2 = 0. T h e nodal curve consists of an equilateral hyperbola in the #jy-plane, through the vertices of the cones. T h e lines riX-\-r2y — ar\=0 and r2x+riy = 0 intersect in a point of x2—y2 — ax = 0, and the lines through this point, parallel to the z — axis is a nodal line. T h e ruled quartic N o . 9 of series I is of the same type as those described under Nos. 20 and 21. T h e y are particular cases of the following kinematic method of generating a certain class of ruled surfaces. Given two right circular cones C\ and C2. A tangent plane e\ revolves around C\ with an angular velocity t\. A tangent plane e2 revolves around C2 with an angular velocity /2. W h e n e\ and e2 are m a d e to correspond to each other according to the law \t\=jxh where X and f are integers, then the locus of the line of intersection g of e\ and e2 x is an algebraic ruled surface. In N o s 9, 20, 21 X = /x. [7]