APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 101 



(8) 



rfZ= 



BX BY BY BY 



BY BZ BY BZ 



Putting now in integral (3) 

 ^_ B o_ 3 9 



BY BZ BY BZ BZ BX BZ BX BX BY BX BY 



which are rational functions of X, Y, Z, and as such admissible as 

 values of a, iS, 7, (3) reduces to 



r dx 



BY'BZ by BZ 

 The analytic form of Abel's theorem is in this case 



x^ 



«, 



BY BZ BY BZ 



BY, BZ, BY, BZ, 



where x„ x„ tCj, . . . , a?^ are the abscissas of the points of intersection 

 of r with the variable surface. 



§7. Application to Quartic in Space of the First Kind. 

 1. The parametric equations 



a;=asni*, y=bcnUj 2=cdnUy (11) 



represent a curve of this kind, since they satisfy the equations 



