104 UNIVERSITY OF COLOKADO STUDIES 



from which 



n— 1 n 



^'^2n-\-l — ^V^-\-fh — U^+,=0, (19) 



hence, from (17) 



Sy^n+i— 2v^ = 2:c,,+,-2c,„, (20) 



which proves the proposition. Hence the theorem : — 



The vertices of a closed jpolygon of 2n sides may he moved in 

 such a manner on the hase-curve of a pencil of quadrics [hyperlo- 

 loids of one sheet) that each of its sides describes an hyperholoid of 

 this pencil,' 2n — 1 of tliese hyperboloids may he chosen at random^ 

 and by these the last one is perfectly determined. 



2. Assume a single hyperholoid passing through T and any 

 generatrix of one of the ruled systems of the hyperholoid defined by 

 Wj+i/j^c; through u^ pass a generatrix u^u^ of the other system so 

 that — (■Wj+'^s) =c; through ii^ pass again a generatrix n^u^ of the 

 first system; and so forth. To find the condition that such a polygon 

 of generatrices closes we have, as before, an even number, and the 

 congruences : — 



Ui-^Ut=Cy 



(21) 



By addition 





Ui—u^^i = 2nc. 

 In order that u^—u^-\-l = we have 



2nc=4:mk-\-4:im^k^ 



2mk 4- 2im,,k. 

 c= ^ L_L. (22) 



n 



