APPLICATIONS OF ELLIFIIC FUNCTIONS TO PROBLEMS OF CLOSURE 89 



arbitrary coefficients remain in C'„ in a linear and homogeneous 

 form. These coefficients may be chosen in such a manner that the 

 curve C'„ = passes through Sn — 1 points on the cubic /"(a?, y) =0, 

 80 that the remaining point of intersection is entirely determined. 

 Hence one and only one relation can exist between the values 

 u^, -Wj, . . . , ^3^ of the parameter u corresponding to the 2u points of 

 intersection. Hence, by Abels theorem 



Wi+'W2+ • • • +Wjj„^ const, (mod. per.) 



5. In case of a conic 



The constant may be neglected without loss of generality. 

 The conic has a contact of the 6th order with the cubic if u^=^ 

 u^^= . . . ^=u^j i. e., if 



3 



Here m and in^ can take all values from to 5, which gives 36 

 points. Among these the 9 points of inflexion are included, so that 

 in reality only 6^ — 3*-= 27 points of contact of hyperosculating conies 

 exist on a cubic. These points are six by six on conies, for example 



w-\-w. 



3 

 2w-\-2w^ 



u„ 



u. 



u. 



3 



3 



' 3~ 

 2w-\-6wi 



3 



the sum of whose arguments is 4:w-{-4:W^. 



