515 



nuinber of points of intersection with an arbitrary straiglit line m', 

 is equal to the number of conies of .S',' tlie planes of which pass 

 tluouf!;h a straiglit line in. From the order just found for kp there 

 follows that tlirough a point P of m there pass six conies of »S, 

 the planes of which contain m. All conies of *S, in planes through 

 m, consequently form a surface which has m as a sextuple straight 

 line and which is of the eighth order, as a plane through in contains 

 one more conic of this surface. Consequently among the conies of 

 «S, intersecting /, there are eicilU the [)lanes of which pass through 

 m and the surface Oi associated to .S',' is accordingly of the eighth 

 order. Evidently the pencil associated to a point of any of the 

 straight lines a^,...,a^ contains a double conic of .S',' and in 6/ 

 tiiere always lies one individual of the pencil of degenerate conies 

 associated to a point of one of the straight lines <',,, . . . , <'j,. 



Tlie si/stem S', formed by the conies cutting a straight line I and 

 a,, . . ., a,., is therefore represented on a surface Oi of the eighth 



order, of which (i\, . . ., a\ are double straiglit lines and t\^, , /j, 



single straight lines. The two tangent planes at a point of one of 

 the straight lines a' \o Oi are the polar planes of the points where 

 the double conic of SJ corresponding to this point, intersects the 

 straight line a associated to a'. 



Finally we investigate the surface Oy which is the image of the 

 system S," of the conies of S, touching a [)lane </. The order of 

 Oy is again etpial to the number of conies of *S," the planes of 

 which pass through an arbitrai'y straight line m. The surface of 

 the eighth order of the conies of which the planes pass through 

 in and which cut a^, . . . , a^, has in common with (/• a curve k' 

 of the eighth order which has a sextuple point in the point of 

 intersection {)n,<p) of m with q. As each of the conies of this surface 

 has in common with k" a pair of points lying on a straight line 

 through {ni, <f>), the number of individuals touching (f is equal to 

 the number of tangents which can be drawn out of (m, 7) to k", 

 i.e. 8 X 7 — 6 X 7 := J4. The system 5," contains consequently four- 

 teen conies the planes of which pass through m and the order of 

 Oy is accordingly fourteen. Now »S'," has two double eonics in the 

 pencil corresponding to a point of one of the five straight lines a' 

 and this system has one individual in common with the pencil of 

 degenerate conies corresponding to a point of one of the lines t'. 

 This individual is u double conic of »S',"- For if we take a straight 

 line m of its plane, it counts twice among the conies of S^' the 

 planes of which pass through vi. The above mentioned pencil of 

 degenerate conies splits off from the system of the conies of 5, 



34* 



