519 



curve of liie order 14, /;", the points of wliicli inay be associated 



luiivaleiitly to tlie conies of 12" passing tlirongli iliem, so that X'* 



13 X 12 

 has the genus /we and accordingly 5^73 double points. 



Six of them lie in each of liie four points of intersection of '/ with 

 one of the lines a^,...,a^ and also there belong to them the four 

 points of intersection outside a, of <( with the double conies of 

 ii'*" the planes of which do not pass throngh a^, and the points of 

 intersection of ip with the six transversals thiongh P of two of the 

 lines rt,, . . . , rtj. Besides these there are 39 more double points. 

 Hence the double curve of 42'* cuts the line a, in 26 points. These 

 are points of a^ throngh which there pass two conies of our system 

 that have there a common tangent plane throngh a,. 



The surface ii" has a twelvefold point in P, as according to 

 ^ 2 our system contains six conies that cut a straight line through 

 P outside P. A plane through P intersects Ü" in a curve of the 

 order eighteen and the genus five as again the points of this curve 

 may be associated univaleiitly to the conies through them. This 

 curve has consequently 131 double points. 66 of them lie in P, 

 six in each of the points of intersection with the lines a, and also 

 the points of intersection outside P of tiie five double conies with 

 the plane must be counted. There remain accordingly 30 double 

 points. 



The double curve of 11'" cuts each line a in 26 points and has 

 in P a 35-fo!d point. 



To the 35 branches of the double curve through P there corre- 

 spond as many pairs of eonics of ii'" touching each oliiei' at this 

 point. Outside P and a^ it must have four more points in common 

 with the plane {P,a^). These lie in the |)oints of intersection outside 

 P of the double conic in the plane [P, a,) and the two straight 

 lines joining P to the points where the transversals of a,, . . . , a^ 

 cut the plane. For these two points of intersection are double points 

 of the curve under consideration. 



Analogously we can examine the double curves of the sui'faee 

 42'* consisting of the conies that cut six given straight lines and 

 the planes of which pass through a given point, and of the surface 

 i2" formed by the conies that cut five given straight lines, touch 

 a given plane, and the planes of which pass through a given point. 



\ 5. We shall first determine the genus of the curve A-'* belonging 

 to the intersection of two surfaces Oi and Oi. The cone A^'* pro- 

 jecting P* out of an arbitrary point K, has in common with Oi 



