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UNIVERSITY OF COLOEADO STUDIES 



of the fourth degree. This is also our idea for the solution of closed 

 linkages. 



The properties of circular series may be extended to space. In 

 this connection I state without proof the theorem': — 



Any closed series of tangent -spheres formed hy generating 

 spheres of one system of a Dupinian cyclide is conjugate to any 

 closed series of the other system. 



If the ratio of the nuimher of revolutions into the number of 



k 



spheres of one closed series is — -, then every series of the other system 



closes, and the ratio— ^ belonging to these series is such that 



m„ 



m, r;^, 2 



The same property holds for circular series and their conjugates 

 in a plane, as Steiner has stated in a celebrated propo8ition\ 



Pig. 10. 



(1) See Annals of Mathematics, Vol. XII, p. 159. 



(2) Loc. cit.. p. 43. 



