156 



SCIENCE 



[jST. S. Vol. XLIII. No. 1101 



remind us of the natural connection. Let 

 the base points be A and B. Choose at ran- 

 dom a point 1 on the cubic curve, and draw 

 in their order the lines A12, B23, J.34, 

 B4:5, etc., all the numbered points lying on 

 the curve. If this series never closes, the 

 same would be true if point 1 were chosen 

 elsewhere ; or if it does close after 2n sides, 

 the same will be true for every position of 

 point 1. Here of course the relation of the 

 base points is decisive, and the fact that 

 elliptic arguments of three points in a line 

 sum up congruent to zero makes the proper 

 choice of the point B a mere matter of 

 arithmetic, i. e., division of a period of the 

 functions for that cubic curve. 



Projection from any point of a twisted 

 quartic curve gives in the plane a cubic 

 curve. But also from one of four spe- 

 cial points, the quartic projects into a 

 conic double. At the same time the gen- 

 erators of any quadric surface containing 

 the quartic curve are projected into tan- 

 gents of a second conic. Any Poncelet 

 polygon of 2w sides on those two conies is 

 then the projection, if we please, of a sys- 

 tem of generators from the two families on 

 the quadric, alternating, n from each 

 family. On the plane cubic the same set of 

 lines would be projected from a point P 

 as the alternating sides of a Steiner poly- 

 gon, where points A and B are projected 

 by the two generators through the point P. 

 As all generators of one family meet every 

 generator of the other family, this makes 

 clear the intimate connection between 

 Steiner's and Poncelet's polygons. 



To vary the object, look at Hurwitz's 

 plane quartic curve with two cusps and a 

 node. It has two inflexions and a double 

 tangent, and is therefore of class 4, dual to 

 itself. On such a curve let a tangent be 

 drawn, and through each intersection with 

 the curve the second possible tangent from 

 that point; we have clearly another (2, 2) 



correspondence, and are prepared for the 

 discovery that closure in a finite number of 

 sides, starting from any one tangent or 

 vertex, implies closure in the sam^e number 

 of sides, whatever the point of beginning. 

 In place of two conies we have here the 

 one quartic, but the essential (2, 2) corre- 

 spondence is in evidence, and the same 

 mobility of figure results from it. 



Not to be confounded with these exam- 

 ples is the particular plane quartic curve in- 

 vestigated by Liiroth, which admits the in- 

 scription of a complete pentagon. There 

 is a resemblance, it is true, in the fact that 

 it too is a problem of closure, and in the 

 variability of the pentagon. For if the 

 sides of one such pentagon are given by 

 equations, p = 0, g = 0, etc., so that the 

 quartic equation is 



pqrs + qrst -{- rstp -\- stpq + tpqr^O, 

 then these five sides are tangents of a 

 unique conic, and every tangent to that 

 conic is one of a set of five constituting an 

 inscribed pentagon of the same quartic. 

 But the correspondence is (4, 4), and the 

 circumscribed locus is not a rational curve. 

 It is, however, in one direct line of descent 

 from Poncelet's triangles. Those triangles 

 mark, on the conic-bearing tangents, sets 

 of three points in involution; and any 

 cubic involution of tangents has for locus 

 of the vertices of its triangles a second 

 conic. So when the number of tangents in 

 each set is increased, we have the involii- 

 tion-curves. It is an involution of the fifth 

 order which generates for its locus this 

 quartic curve of Liiroth, each tangent inter- 

 secting the four in its own set. Such an 

 involution is the equivalent of a (4, 4) 

 correspondence, which might in special 

 cases degenerate into two (2, 2) corre- 

 spondences, and carry Liiroth 's quartic 

 curve with it into two distinct conies, each 

 containing a system of inscribed Poncelet 

 triangles. 



