516 



cutting m twice, so that there remains a surface of the seventh 

 order wliich intersects <f along a curve F with a fivefoki point in 

 [in, If). Instead of 14 we can now draw 7 X 6 — 5 X 6 = 12 tang- 

 ents out of {m, (f>) to this curve. Hence a straight line »(' through a 

 point of a straight line t' has in this point two coinciding points of 

 intersection with Oy. 



The system S'\ formed by the conies of »S, touching a plane <p, 

 is represented on n surface 0-^ of the fourteenth order of which 

 a\, . . ., a\ are Quadruple straight lines and t\^ , t\, double lines. 



^ 3. From the investigated representation we can now in the first 

 place derive the number of conies which cut five straight lines and 

 which fulfil a threefold condition '). 



2 X 2 =: 4 of the 48 points which a curve kp has in common 

 with a surface O, fall in each of the double points of kp and one 

 in each of the ten points of intersection of kp with the straight 

 lines t'. Accordingly the curve kp cuts a surface Oi in eighteen 

 points which are not singular for the representation. 



There are therefore eighteeii conies passing through a given point 

 and iiitersecting si.v given straight lines. 



2X4 = 8 of the 84 points in which a curve kp intersects a 

 surface 0^, lie in each of the five double points of kp and two in 

 each of the ten points of intersection of kp and the lines t' . Here 

 we have therefore 24 points of intersection that are not singular 

 for our representation. 



There are accordingly 24 conies passiiig through a given point, 

 touching a given plane and intersecting pre given straight lines. 



Of the curve of the order 64, which two surfaces Oi have in 

 common, each of the straight lines a' splits off" four times and each 

 of the lines t' once. There remains, accoidinglj', a curve of the 

 order 34, k^\ which is the representation of the system of the conies 

 in S, cutting two given straight lines. The conies of this system of 

 which the planes pass through an arbitrary point, are represented 

 on the points of intersection of k'* with the polar plane of this point. 



There are therefore 34 conies wlncli cut seven given straight lines 

 and of lohich the planes pa.ss through a given point. 



We have found in § 2 that there are eight conies which cut six 

 given straight lines and of which the planes pass through a likewise 

 given straight line. Hence the system associated to X" contains eight 



') Gf. Schubert: „Kalkül der Abzahlenden Geometrie", p. 95. 

 Jan de Vries, These Proceedings, Vol. IV, p. 181. 



