421 



This number is evidently the order of the twisted onrve *!*, which 

 has / as image. 



As / has apart from the principal points, a number of points in 

 common with (7*"+6(.4|"+3, Gk^), represented bj 



n* = {2n + 3) {2p — ^at) — 2:Sgk, 



the curve <P rests in ?i* points on the curve «'. A straight line / 

 is therefore the image of a a'-"+^, which intersects «' in (4?i-}-6) points. 



6. For the simplification of the representation we submit the 

 figures in T to a quadratic transformation, which has Jjt as principal 

 points. By this the curve c-"+"' is transformed into a curve c"-\-- , 

 which does not pass through .4,',-, but does pass through the (/i^-)-2/i-|-3) 

 points @, in which the principal points G are transformed. 



To the curves 7-"+^, in which the surface *?»2"+i is intersected by 

 the planes y of a pencil, correspond now the curves of a pencil 

 (c"+2). Among them there are 3(n-\-lY, which possess a nodal point, 

 which is then at the same time the case with the con-esponding 

 curves y2"+i. 



The surface ^^.i+i (,. consequently of class 3 {ii-\-\y. 



The straight line ©j t^V^ is transformed by the quadratic trans- 

 formation into the conic ƒ^(^^ G^G^) and the latter is the image 

 of a twisted curve «i>", which rests on a^ in {2n — 1) points. For 

 through ƒ- and «' passes a hyperboloid, which has the curve ƒ'' and 

 n times the curve u^ in common with <p2-i+i . ^\yQ residual section 

 is the '^" in question. 



7. We shall now consider a sur j ace 0"+7^+i that passes n times 

 through a twisted curve a^i of order q, and p times through a straight 

 line ^, lohich is {q — 1) times intersected hy a'l. 



The straight lines t, which intersect a and ji, form a linear 

 congruence (1,^'), by which is represented in a plane t; for t 

 intersects 4>, except on a and /?, only in one point P more. 



The ruled surface ^, which represents the plane section y"+/'+i is 

 of order {n-\-p-\-q-\-4.); for in the plane y lie q generators. 



Out of a point of i3 the curve «9 is projected by a cone of order 

 q with ((7+I) fold edge ;:?, which intersects y"+/^+i in {p-]-q) 

 points P. Consequently (i passes {p-\-q) times through /?. 



Out of a point of « the line 3 is projected by a plane that deter, 

 mines {n-\-l) points P on y. Consequently o7 is an (w-|-l) fold directrix. 



The image of y"+/^+i is consequently a curve c'^+z^+^+i (qA"+^, Bp+9)^ 



Two curves c have apart from the points A and B and the images 

 of the (n-fp+l) points in the section of their planes a number of 

 points G in common represented by 



