( 838 ) 



This is at the same time the genus of (>'". It is evident tliat p 

 may not be smaller than {^q -f- 1). For the smallest values of p 

 and q we have 



The above considerations mav be extended b}' taking instead of 

 scroll 0^ a scroll $« with (n — l)-fold right line d. Out of a point 

 of (/ now start {?i — 1) right lines a^, a^, . . . a,,—! . A (p, q)-cor- 

 respondence between the pencils of planes [a^) and (d) determines 

 again a twisted curve of order j) -\- q = m, having as central projec- 

 tion a C" with ^;-fold point D and (/-fold points in A^, A^, ... An— \ ; 

 and inversely q^ is again entirely determined by C" . For the genus 

 of ^"' (and c"") Ave now find 



9 = -J (P 'h q - '^) ip + q - ^) 



P{p 



^)-^qiq 



1) (n - 1) 



or 



9 = {q 



1) {m - 1) _ _ J (^ _ 1) n. 



To obtain general twisted curves we shall not be allowed to take ;; 

 larger than 4, q larger than 3. For n = 2 we find evidently the well 

 known coiisiderations concerning curves on an hyperboloid. 



5. If we substitute in t for the curve c" a curve passing p times 

 through D, q times through Ai- and moreover cutting the right line 

 DAk in s points, then this curve is evidently the central projection 

 of a curve on 0'\ having a multiple point in 0. For, each of the 

 {n — 1) tangential planes in will now contain s right lines, touching 

 the twisted curve in 0. 



