APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 123 



Bj, B,; B^, B^; B^, Bj ... respectively. In a closed link- work this 

 series of circles closes also, so that the last point of intersection B^_^, 

 will coincide with the first point B^. This result may be stated in 

 the following form : — 



If two fixed circles A and B are given, a series of circles can be 

 drawn, whose centres AjAjA.^, ... all lie on the circle A and which 

 all pass through the centre O of A. The first circle Aj of this series 

 intersects circle B in two points B^. The second circle A^ passes 

 through Bj and intersects circle B a second time in Bg. The third 

 circle passes through B2 and intersects B a second time in B^, and so 

 forth. In this manner a series of circles is obtained which may be 

 divided into three different classes : — 



I. The series is limited, i. e., the construction cannot be con- 

 tinued indefinitely. 



II. The series closes, i. e., after the construction of a certain 

 number of circles, the last point of intersection B„_|., will coincide 

 with the first B,. 



III. The series is unlimited. 



According to the general theorem on the link -work it follows 

 immediately that if the series of circles closes once, it will close in 

 all cases, no matter where the first circle of the series is drawn. If 

 the series does not close in one case, it never will close. 



The circles of the previous series all touch a circle C of centre 

 O and radius 2r. Applying to this series an inversion with centre O 

 and any radius, every circle of the series is transformed into a straight 

 line segment, tangent to the transformed circle of C and inscribed to 

 the transformed circle of A. Thus the series hecomes a polygon 

 lohich is inscribed to one and circumscribed to the other circle. 

 This is precisely the case of Poncelefs polygons^ Fig. 9.^ As to the 

 properties of closing of these polygons, it is evident that they are the 

 same as in our link-work and the series of circles derived from it. 

 The system of circles from which Poncelet's polygons arise may also 



(J) In Poncelet's Traite des proprittea projectives des figures (1822) §565. See also Greenhill, 



Elliptic Functions, pp. 121-130. 

 (^) In Fie. 9. C has been chosen as circle of inrersion. 



