( 837 ) 



Mathematics. — ''On algebraic twisted curves on scrolls of order 

 n ivith {n—lyfold right line." By Prof. Jan de Vries. 



1. If we intersect a, cubic scroll 4>' b}' a pencil of planes having 

 a generatrix a of <P^ as axis, we get a system of conies (>*, all passing 

 through the point 0, where a meets the double right line d. If we 

 take a {p, g)-correspondence between this pencil of planes and the 

 pencil of planes with axis d, then in this way to each (.." are 

 assigned p right lines r of 4'' and to each right line r evidently q 

 conies jj\ The locus of the points of intersection of the lines ?' and 9" 

 corresponding to each other is a twisted curve of order m:=p-j-^; 

 for the points of the rational cubic curve whicii 0" determines on 

 an arbitrary plane are arranged in a {p, 5')-correspondence, of which 

 each coincidence is the point of intersection of a 9" with a right 

 line r corresponding to it. 



The twisted curve 9'" has the right lines r as ^-fold secants, 

 whilst it is intersected by each of the co^ conies of 0'^ in p points. 



2. If 4)^ is represented by central projection out of on a plane 

 T cutting a, and d in A and D, then the systems (?') and (p"") are 

 transformed into the pencils (Z)) and {A) which are now likewise 

 arranged in a (/;, 5')-correspondence. The curve C" generated in this 

 wa}^ has in Z) a />fold point, in A a 5-fold one. But it has moreover 

 a ^--fold point in the point of intersection B of the right line b of 

 <P'^, which still passes through 0; for b is g'-fold secant of 9'" . 

 From this ensues that the correspondence {p, q) in t cannot be taken 

 arbitrarily. 



The curve ^"' is completely determined by its central projection 

 c'" . For, the cone projecting c'" out of has a ^9-fold edge along 

 d and g-fold edges along a and b; so its section with (p"' consists 

 of 2p -\- 2q z= lin right lines and a twisted curve of oider m having 

 p points in common with d and q points with a. 



As the singular points of C" are equivalent to hp {p~A) -j- q {q — 1) 

 nodes, the genus of C" is indicated bj 



1 1 



2 



= (p-i)(^-i)-ir^(^-i), 



or by 



g = {m-l){q-l)-~q{q-iy 



