518 



planes of which pass through /«,, which cut r/,, a^ and r/,, and 

 which touch y, and rp,, and this surface is of tlie 24''' order. The 

 number of conies in question is therefore '24, and k'" lias 24 double 

 points on each of the Hues n'. As for instance the line /,, is not a 

 component part of any degenerate conic cutting r/,,...,ö'j and 

 touching fpj and /p^, X" has no point in common with awy of the 

 lines i'. 



If we now determine the numbers of points of intersection of 

 k'' with surfaces Oi and 0,^ that are not singular for the represen- 

 tation, we find resp. : 



There are 128 conies intersecting sir given straight Hi^es and touching 

 two given planes. 



There are 104 conies intersecting jive given straight lines and touching 

 three given planes. 



^ 4. The genus of the system of conies through a given point 

 P intersecting ^/j, . . . , a^, is equal to that of the associated curve 

 kp, which is of the sixth order and has five double points; conse- 

 quently it is five. According to the first theorem of ^ 3 these conies 

 form a surface of the eighteenth order, ii'". To a conic of 52'* we 

 associate the two points in which it intersects a |)lane fp which therefoi'e 

 always belong to the curve X'" along which i2'" is cut by 'p. To 

 the (l,2)-correspondence between the conies of 52'* and the points 

 of X'" arising in this way, we apply the formula of Zeuthen : 



'J,— »J. = 2<(, (/>,— 1) — 2«,(/j,— 1) (1) 



In this case «j = I, «, = 2, />, = 5, ijj = S,nd 7j, = the number 

 of conies of 52" toneliing (p, that is, according to ^ 3, 24. By sub- 

 stituting these values in (1) we find that p„ i. e. the genus of X", 

 is equal to 21. Hence the curve k^" has 115 double points. Among 

 these each of the points of intersection of '/' with a line a in which 

 k^" has quadruple points, must be counted six times. Further there 

 belong to them the ten points of intersection of (p with the five 

 double conies of i2" of which the |ilanes pass through one of the 

 lines a, and the ten points where q is cut by the double straight 

 lines of 52" i.e. the transversals tp of two of the lines « which 

 pass through P and form a conic of 52'" together with' the two 

 transversals of tp and the three remaining lines a. There remain 

 accordingly 65 double points. 



The surface of the conies through a given point which cut Jive 

 straight lines, has a double curve of the 65''i order. 



A plane rp through a, has in common with 52" besides a, a 



