APPLICATIONS OF ELLIPTIC FUNCTIONS TO 

 PROBLEMS OF CLOSURE 



Bt Arnold Emoh 



INTRODUCTION 



Elliptic functions as a special branch of the theory of functions 

 occupy an extremely important place in modern mathematics. Sim- 

 ilar interest is attached to them with reference to their beautiful ap- 

 plications in geometry and mechanics. As a result of the investiga- 

 tions along these lines we have the valuable treatises by Gkeenhill, 

 Halphen, Appell and Lacour, and a great number of monographs 

 distributed in various mathematical journals. Other treatises on 

 elliptic functions, with a few exceptions, invariably contain a 

 chapter on the applications. 



In this article I shall make a limited collection of problems 

 concerning variable figures with the property of closing in every 

 particular position. 



Naturally many of the problems to be treated are well known in 

 one or another form, while on the other hand, in the last two chapters, 

 I shall add some of my own investigations on this subject, partly 

 new, partly already published. 



The first chapter treats of Abel's theorem and its application to 

 plane curves of the third order. Here the principal object is to bring 

 out Steiner's celebrated problems of closure (Schliessungsprobleme) 

 by the method first established by Clebsch. 



The second chapter contains a generalization of Abel's theorem 

 and its special application to twisted curves of the fourth order of the 

 first kind and the pencil of quadrics passing through it. A sketch 

 of the constructive treatment of the same problems concludes the 

 chapter. 



