520 



besides k'* a curve of tlie order 238, P". Tins curve lias double 

 points in each of the double points of k'\ because the enlire inter- 

 section of /v'* and Oi has a quadruple point in such a point. Further 

 K'* cuts each of the lines n' in 18 more points and here F'* has 

 double points. But this curve has 32 single points on each of the 

 lines t'. 



The surface 0/ is cut by k"" in 



238 X 8 — 5 X 26 X 4 — 10 X 32 = 1064 

 points that are not singular for the representation. These are points 

 of intersection of /" and k^*^ ; a part of them lie in the points 

 where a generatrix of /v" touches the surface Oi, hence in the 

 points of intersection of k^* with the polar plane of K relative to 

 0; that are not singular for the representation. As this polar surface 

 is of the order seven and passes singly through the lines a', it cuts 

 k'* in 7X34 — 5 X 2 X 8^158 non-singular points. The remaining 

 906 points of intersection of k'* and k'"^ are the points where the 

 bisecants of k'* through K cut this curve. Hence there pass 453 



34X33 



bisecants of k" through K, and in a plane there lie — =r561 



A 



bisecants of this curve. 



According!}' : 



The bitaiifients of the developable surface that is enveloped by the 

 planes of the conies intersecting seven given straight lines, form a 

 congruence (561, 453j. 



As K** has 453 -j- 5 X 8 = 493 double generatrices, the genus 

 of the curve k'\ hence also the genus of tiie system of (he conies 

 cutting the lines r/,, . . . ,a^, I and /', is equal to; 16 X 33 — 493r=35. 



To each conic of the surface ii." corresponding to the curve k'*, 

 we associate again the pair of points in which such a conic cuts 

 an arbitrary plane ^f■, it belongs to the curve k^'' along which i2" 

 intersects the plane (f. We apply the formula: 



>i,— -»i, = 2«, (/>,— 1) — 2«, (p— Ij . . . . (1) 

 to the correspondence (1,2) arising in this way between the conies 

 of i2" and the points of k". Here »jj ^ the number of conies cutting 

 seven straight lines and touching a plane; according to § 3 it is 

 116. Further tj, = 0, «,^1, «, = 2 and p, = 35. By the aid of 

 these values there follows from (1) that the genus of ^" is equal 

 to 127. 



The number of double points of ^" is consequently 91 X 45 — 127 := 

 = 3968. As there pass eighteen conies of ^" through a point of 

 one of the directrices of this surface, whence these directrices are 



i 



