( i'^^ ) 



The two theorems mentioned above and their reciprocals and some 

 special cases will now be treated algebraically. 



§ 4. Let 6'i be a rational twisted curve of rank /j and ^S' a surface 

 of order n, possessing no «nnltiple curves. Let 



ax -\- by -\- cz -\- d zzi 



represent the osculating plane of C^, a, h, c, d being integer rational 

 algebraic functions of t. Differentiating we find for an arbitrary tangent 

 of Ci equations of the form 



«1 •^' + ''i ^ + Cj c + d^ — 0, 



«2 - + i\ y -\- c^z -\- d^ — Q. 



Solving y and z in function of x and t we find : 



Aw + B Dx -I- E 



y^ — ^^'"^ — c~' (^) 



in which A, B, C, I) and E are functions in t of order i\. If we 

 substitute the values (.1) in the equation of the surface S, we arrive 

 at an equation (7i), which is in x of oi'der n and in t of order 

 n i\. For every value of t this equation {B) furnishes the n values 

 of X belonging to the points of intersection of a tangent I Xo C . 

 If two of these values become equal, the tangejit / will meet the 

 surface >S' in two consecutive points and as .S' is supposed to have 

 no multiple curves the tangent / will also be a tangent of>S'. Those 

 tangents of C, are excluded which are at right angles with the A'- 

 axis, all points of intersection with S possessing the same x; so all 

 roots ,1' coincide, without the points of intersection coinciding. Everv 

 line being at right angles with the A'-axis meets the line at infinity in 

 the plane d == 0. So the number of these particular tangents of Cj \ïh\. 



The equation (7i) has two equal roots in x for a certain value of 

 t, when this value of / causes the discriminant of (7i) to vanish. 

 The discriminant is in the coefficients of [B) of order 2 {n — 1) and as 

 the coefficients of (7J) are of order i\ n 'm t, the discriminant is of 

 order 2 }\ n {n — 1) in t. 



By a parallel displacement of the axes the plane x = can be 

 made to i>ass through one of the tangents of C^ which is at right 

 angles witli the A-axis. 



Writing t -\- q for t, we can take q in such a way that this 

 tangent of C\ lying in x = corresponds to the value / = 0. The 

 equation (B) has then passed into an ecpiation (B'. where foi- / = 

 all roots X \'anish. 



The first equation (A) 



