rEBBnAKY 4, 1916] 



SCIENCE 



153 



case of two circles. Steiner had but re- 

 cently written on the same topic, appar- 

 ently unaware that it had been approached 

 before. Jacobi was able now in the light of 

 Poneelet's theorem to vindicate the claims 

 of Fuss to priority, since his irregular- 

 symmetric polygons were particular cases 

 in every infinite set of Poncelet polygons 

 on the same pair of circles. Jacobi further 

 applies the recursion formulae arising in 

 the iterated addition of a constant to the 

 elliptic integral of the first kind. Note his 

 compact and expressive formulae. If the 

 radii are R and r, the distance of their 

 centers a, and the w-gon encircles the cen- 

 ters i times before closing, all this is duly 

 contained in the three formulte 



s: 



1/(1 — khsYc? ip) 



-.ir 



„ Jo 

 r 



4:aR 



-/(I — kksiD?<p) ' 



{B+ a)' - rr " 



By this apparatus he verified the condi- 

 tions already calculated for the closure in 

 3, 4, 5, 6, 7 sides, and confirmed for 8 sides 

 the result of Fuss in opposition to Steiner 's 

 formula. 



Certainly there is something satisfactory 

 in seeing similar steps in geometric con- 

 struction replaced by successive additions 

 of one fixed quantity to an elliptic argu- 

 ment. But the problem was originally one 

 of algebraic geometry, in so far as the 

 conic represents a quadric form and the 

 conditions of incidence and contact are 

 algebraic ; hence it was to be expected that 

 there would be investigators who would not 

 be satisfied with this transcendental eluci- 

 dation of Jacobi, but would insist upon 

 algebraic treatment throughout. More- 

 over, when once the projective treatment of 

 figures had acquired prestige in pure geom- 

 etry, it made inroads rapidly in the analytic 

 territory. It was then desirable to solve 



the problem in its generality, for two conies 

 whose equations are given arbitrarily, not 

 restricting them to be circles; and to use 

 processes and nomenclature that would not 

 be affected by linear substitutions upon the 

 coordinates or coUineation. These last two 

 desiderata appealed to Cayley not long 

 after 1850, and from time to time he worked 

 out parts of the problem: to express in 

 invariants of two quadrics the condition 

 that a broken line inscribed in the one and 

 circumscribed to the other shall close in n 

 sides. The results are not stated in terms 

 of rational invariants, but they have the 

 very great merit of being quickly and easily 

 perceived, and of requiring only invari- 

 ant terminology. The discriminant of a 

 quadric is perhaps the best known of all 

 invariants. For a quadric with one linear 

 parameter he requires the discriminant to 

 be calculated — namely for F -\-K(^, where 

 i^ = and $ = are the equations of the 

 two conies, respectively. This is of degree 

 3 in the parameter. 



Next, the square root of this discriminant 

 is developed formally in ascending powers 

 oiK; 



V'S=VA+ B^K + B,B? -I- CiS^ 



C^K<- + D^K' + D^K' + etc. 



The conditions of closure are now, in 

 form at least, simplicity itself, namely, the 

 vanishing of a determinant whose consti- 

 tuents are coefiScients in this development. 

 For an odd number of sides in the polygon, 

 the leading constituent is C^; for an even 

 number, C^, thus: 



For 3 sides. 



