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SCIENCE 



[N. S. Vol. XLIII. No. 1101 



= 0; 



For 4 sides. 



For 6 sides, 



For 8 sides, 



C2 A I 

 A A 



Ci A A I 



Di A ^i = 0, etc. 



A E\ E2 1 



When one of these conditions is satisfied, 

 the corresponding polygons are inscribed in 

 the conic $ = 0, and circumscribed to the 

 other. To test two given conies by this 

 method would evidently involve consider- 

 able labor, but it would have the merit of 

 being straightforward work, all of one 

 kind — ^the calculation of determinants. 

 Only one such would enter, the square root 

 of the discriminant of the conic that carries 

 the tangents, hence rationalization would 

 be easy. It is hardly likely that results 

 more elegant will be reached by any 

 method ; yet there are later researches, that 

 I have not yet been able to examine, highly 

 praised by reviewers. It does not appear 

 that Cayley has given any account of the 

 modifications necessary in these conditions 

 when the sides touch different curves of the 

 pencil. 



Two other questions, however, were 

 started by Cayley. The first is that of the 

 relations in terms of the two invariant 

 cross-ratios of the two conies — those be- 

 longing to the four common points or the 

 four common tangents in the one conic and 

 in the other. Conditions that exhibit a re- 

 cursive law of formation in one domain of 

 rationality are quite certain to do the same 

 in a different domain, and Halphen has 

 carried out the solution of this problem to 

 completion (if that is a possibility) in his 

 Elliptic Functions, Part 2. His interest 



in the geometry of the figure led him to 

 propose the question, How many conies in 

 a linear system can serve as loci for the 

 vertices of a polygon of m sides, the sides 

 to be tangent to a fixed conic 1 The answer 

 is, for a polygon of 3 sides, 2 conies; for 

 5 sides, 6 conies; for 6 sides, 6 conies; 

 in general 



?(■-?)('-?)('-.)-. 



where all the prime factors of m are p, 

 q, r, etc. 



Cayley 's second new problem in this con- 

 nection was one concerning curves other 

 than conies. If m^ denotes the order; /*» 

 the class of any curve, and it is required to 

 describe a closed polygon beginning from 

 a vertex A upon a curve of order m^, draw- 

 ing a side that shall touch a curve of class 

 /Ai and meet in a second vertex a curve of 

 order m^, and so on, then the number of 

 solutions is twice the continued product of 

 the m's and the ju's. This implies that the 

 curves are all different, and calls for modi- 

 fication when coincidences are required. 



Cayley initiated, but Hurwitz carried to 

 completion, an algebraic explanation of the 

 mobility of the Poneelet polygon whenever 

 it actually exists. This, which is much the 

 simplest method of attack, is by means of 

 a correspondence upon a rational curve or 

 line. The conic is a rational curve, and 

 its points or its tangents can be given by 

 quadric functions of a single parameter. 

 In the presence of a second conic to carry 

 the tangents, any point of the first corre- 

 sponds primarily to two others, namely 

 those two points in which the first conic is 

 cut again by tangents to the second conic 

 drawn from the first point. Such a corre- 

 spondence is symmetrical two-to-two or 

 (2, 2). Points further removed from any 

 given first point are related to it second- 

 arily, or more remotely, by a derivative 



