APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 125 



and tangency). The properties of Steiner's circular series are there- 

 fore not changed by such a transformation. 



From this standpoint it is an easy matter to prove the following 

 theorem: If in a coaxial system of circles A^, A^, A^, . . . , A^, • • • ? 

 with real limiting points, a system of circles B^, B^, B^, . . . , B^ can 

 he described each of which being tangent to any two circles A, and 

 so that Bi cuts a fixed circle A^ of the coaxial system in P^ and jPj, 

 B^ cuts A^ in P^ and P^, B^ in P^ and P^, . . . , finally B^ cuts 

 Ay^ in P^ and P^, then, under the same conditions, an infinite 

 number of closed systems of circles B can be described. In all 

 these closed systems the points of intersection of corresponding 

 consecutive circles lie in circles of the same coaxial system,. 



In reality, an inversion having one real limiting point as the 

 centre of inversion transforms the system (A) into a system of con- 

 centric circles, and in connection with this the theorem is evident. 



If the circles B all pass through one real limiting point, being 

 thus tangent only once to circles of (A), and making an inversion, 

 Poncelet's theorem of the polygon inscribed to a circle and tangent 

 to circles of a concentric system is obtained. By considerations 

 similar to those in connection with linkages the theorem is deduced: — 



If a system of circles B pass through any fixed point Q and if 

 each circle B is tangent to a circle of a coaxial system (^) and cuts 

 the preceding and consecutive circle of [B) in points of a fixed circle 

 of i^A'), then there are an infinite number of such closed systems {B). 



From an inversion with Q as a centre results Poncelet's general 

 theorem concerning closed polygons inscribed to a fixed circle and 

 tangent to circles of a coaxial systems. 



2. The algebraic properties of all these configurations have 

 been studied by A. Hurwitz,^ who has shown that they rest upon the 

 existejice of more than n roots of an equation of degree n. 



Cayley in a number of articles'^ reduces all problems of this kind 



/jOQ ft It 



to the differential equation — ^- — ^ , f being a polynomial 



^A^) ^Ay) 



(1 ) Mathematische Annalen, Vol. 15, pp. 8-15, and Vol. 19, pp. 56-66. 



(2) See in particular, "On thePorism of the In- and Circumscribed Polyffon," etc., Cayley's 



collected Mathematical Papers, Vol. VIII, pp. 14-21. 

 4 



