APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 115 



describe a series of circles. From the figure it is seen that the two 

 circles having A; and A^^, as centres pass through the points B. and 

 B/. The properties of closing of these series are evidently the same 

 as those of the link-work. All circles of the series are tangent to 

 two concentric circles. The figure can be generalized by an inver- 

 sion and we have immediately the result: — 



If each pair of consecutive circles of a series of circles^ which 

 all touch two fixed circles C^ and C^^ intersect in two points B^ and 

 B^' , and if the points B^, B^, B^, , . . are situated on a circle 

 C^, then the points B^' , B^' , B^' , . . . are also situated on a circle 

 O.' (Fig. 6). 



Fie. 6. 



(*) This statement may be seneralized in such a manner that instead of a circle Cs any cnrve 

 is assumed. From Fig. 1 it can easily be proved that in this case Bi', Bj', B3', . . . are 

 situated on a curve which is the inverse of the first with regard to the center O. The 

 special case above has been formulated in view of its subsequent application. 



