Febbuaby 4, 1916] 



SCIENCE 



157 



A somewhat different kind of curve aris- 

 ing from a (2, 2) correspondence was that 

 investigated some years ago by Holgate. 

 Starting with a pencil of conies, normalized 

 to a system of coaxial circles, he gave to 

 every point in their plane an index, usually 

 00, but for certain points finite. From the 

 point any line is drawn. It touches two 

 circles of the system. One of these has a 

 second tangent through that same point, 

 and that tangent touches a third circle, etc. 

 If after n steps of this kind the first line is 

 reached again, the index of that point is n. 

 Holgate determined the locus of points 

 whose index is 3 as a parabola; that for 

 index 4 as a nodal quartic, and laid out 

 the general method for higher indices. 

 One should react from this experiment to 

 something more like the original Poncelet 

 object; to one fixed conic as support of its 

 tangents, and a double infinity or net of 

 conies. A simple infinity of conies in this 

 net would contain Poncelet triangles with 

 respect to the fixed conic : their index would 

 be 3, and their envelope would take the 

 place of Holgate 's parabola. And for the 

 dual problem, there is ready at hand the 

 well-known system of confocal conies, in 

 which the indices of all straight lines should 

 be studied, and the envelopes of lines for 

 each integral index. 



The number of different treatments of 

 this same problem increases, not rapidly, 

 but steadily ; its fascination is exerted upon 

 the successive generations of mathemati- 

 cians, and some of their works of art stand 

 out from the mass, some for a little time, 

 some longer. I shall pass over most of 

 them, these images, in geometric shape, of 

 the algebraic (2, 2) correspondence; and 

 describe only one more related object, an 

 image of a (3, 3) correspondence. Franz 

 Meyer studied it and elaborated it in detail, 

 years ago as a docent at Tiibingen, in his 

 book on Apolaritat. Studying the quartic 



involution, he began with the (3, 3) corre- 

 spondence among points upon a twisted 

 cubic curve, the simplest rational curve in 

 space of three dimensions. For compari- 

 son, remember the cubic involution on a 

 conic in two-space. There we had this 

 theorem on Poncelet triangles: If a conic 

 be circumscribed to one triangle which is 

 circumscribed about a fixed conic, then 

 there are oo^ other triangles similarly re- 

 lated to the two conies. Meyer found the 

 theorem, surprising by contrast: If a tet- 

 rahedron be formed of four planes which 

 osculate one fixed cubic curve in three- 

 space, and a second cubic curve be passed 

 through its four vertices, then that pair of 

 cubics may have, or may not have, a second 

 tetrahedron similarly related to them. If, 

 however, there is a second tetrahedron, then 

 there is a simply infinite set of such. Many 

 other remarkable facts in the geometry of 

 twisted cubic curves he developed, most of 

 which still wait for diffusion among the 

 geometric public. 



Such a discrepancy between conic and 

 cubic does not exist in regard to periodic 

 sets of lines and planes, respectively, of 

 period seven. Whether it is found for pe- 

 riods five and six, no one has yet under- 

 taken to determine. Yet a cleavage so 

 marked, and so unexpected, is certainly a 

 challenge to geometricians to explore fur- 

 ther the so-called norm curves of hyper- 

 space, and the involutions of point sets of 

 low orders upon them. 



Also the half-forgotten fact deserves 

 recognition and exploitation, that all those 

 Poncelet systems are associated with linear 

 involutions upon rational curves. In that 

 feature, possibly, lies even more promise of 

 generalizations and discoveries than in 

 Jacobi's brilliant and beautiful depiction 

 by the aid of periodic functions. 



Not every creation of the geometric mind 

 finds an environment ready in which it can 



