730 



Let ;S be a point of the singular curve 6; the ray FS is cut in 5 

 5y the (?z+3)(?z--l)' curves yc", of which the planes pass through i^>S. 



In connection with what was mentioned above we may therefore 

 conclude that the singular curve ö is (?ï4-3) (7Z— l)'-fold on the 

 surface A. 



5. If all y" pass through the pole F, so that the latter is a 

 fundamental point of the congruence, then all surfaces ^" + i have 

 a node in F. Two surfat^es have four points in F in common in 

 that case; one of tliem belongs to the y", which forms part of the 

 intersection, consequently the lingular curve o has now a triple point 

 in F. In an arbitrary plane (p passing through F the two 2 have 

 {ii-\--\y — 4 points in common, apart from F, {n — 1) of those 

 points lie on the common y", the remaining (yi'-l-?z— 2) on o. 



In those points o is cut by the curve of the congruence lying in 

 (f. The curves y" consequently pass through the triple point of the 

 singular curve, and rest moreover in (n+2)(?z~l) other points on it. 



Any plane passing through a tangent tk in i^ to ö contains a y», 

 which touches tk in F. In the plane passing through two of those 

 tangents lies therefore a yj", wiiich has a node in F. Each of the 

 three bitangent planes of «J which are determined by the three 

 tangents in 7^^ contains therefore a yo^" with node F. 



The quadric cones of contact in F of the surfaces of the net 

 [^"+^J form apparently a net which has as base edges the three 

 tangents of the singular curve o. To that net belongs the figure 

 consisting of the plane tk ti with an arbitrary plane passing through 

 t,n; so the net [^" + i] contains three systems of surfaces, which 

 have a biplanar point in F; the edge of the pair of planes into 

 which the cone of contact degenerates lies in one of the three 

 planes tk ti . 



6. We shall now consider a triply infinite system of plane algebraic 

 curves y«, which form a bilinear complex |y"| '). In an arbitrary 

 plane lies therefore one y", and the curves y", which pass through 

 a point F, lie in the planes of a peiicil (cone of the first class) ; 

 the axis p of that pencil we shall call for the sake of brevity, the 

 axis of P. 



The curves of |y"|, of which the planes pass through an arbitrary 

 straight line r form apparently a surface of order {n~\-\), which we 



1) The bilinear complexes of conies have been fully treated by D. Montesano 

 ("1 complessi bilineari di coniclic nello spazio", Atti R. Ace. NapoH, XV, ser. 

 2a, nO. 8). 



