82 UNIVERSITY OF COLOKADO STUDIES 



In the third chapter I shall show how elliptic functions make 

 their appearance in the theory of certain linkages. Poncelet's poly- 

 gons and Steiner's circular series result from this theory as special 

 eases. 



Loxodromics on the torus, and in general on Dupin's cyclides 

 form the subject of the fourth and last chapter. 



In the first two chapters I started from the most general prop- 

 ositions and passed through the intermediate steps which are neces- 

 sary to show in a connected manner how the Abelian theorem enters 

 into the theory of elliptic functions and their geometrical applications. 



I. ABEL'S THEOREM. APPLICATION OF ELLIPTIC FUNCTIONS TO 

 PLANE CUBICS. PROBLEMS OF CLOSURE 



^1. Abel's Theorem'. 



1. Let 



/(«',y)=0 (1) 



be the equation of an algebraic curve Cm of the wth order and 



any Abelian integral attached to this curve. Consider a system of 

 algebraic curves 



</> K y, <3^n ««•••, «r)=0 (3) 



of the n\h order depending upon r arbitrary parameters «„ a^^ . . . a^ 

 which are supposed to be contained rationally in (3). The curve (1) 

 and each curve of (3) have ^^=inn points 



(a-p yi\ («*« y^y " • y {^fi^ yfi) 



(') For a complete treatment of this theorem see Picard: Trait6 d'Analyse, Vol. II, pp^ 

 393-396. Appell et Coursat: Th6orie des Fonctions Alg6briques. pp. 401-403, which I 

 have here followed more or less. 



