517 



double conies the planes of wliicli pass tliroiigli one of the lines a, 

 and accordingly k*' has eight donble points on each line a'. 



Likewise the system corresponding to k'^ contains pairs of degene- 

 rate conies of which the image points lie on one of the lines /'. 

 For instance to points of t\, there are associated tiie two conies 

 consisting of t^^ and tlie transversals of i,,, a, and the two directrices 

 outside the lines a ours of the sj'stem of conies under consideration. 

 Hence k'* cuts each of the lines t' in two points. 



The curve k'^ cuts a third surface Oi in 272 points. Four of 

 these lie in each of the 40 double points of k'\ and 20 belong to 

 the straight lines t. There are consequently 92 points of intersection 

 that are not singular for the representation. 



There are 92 conies intersecting eight given straight lines. 



From the lunnber of points of intersection of k*'' with a surface 

 Oy that are not singular for the representation, there follows: 



There are 116 conies intersecting seven given straight tines and 

 touching a given plane. 



A suifaee 0/ and a surface 0^ have an intersection of the order 

 112. From this each of the straight lines a' splits off eight times 

 and each of the lines t' twice. There remains a curve of the order 52. 



There are 52 conies which cut six given straight lines, touch a 

 given plane, and of inh'ich the planes pass through a given point. 



Let us investigate the intersection of two surfaces öy more closely. 

 It is of the order 196; each of the straight lines a' splits off sixteen 

 times, each line t' four times. There remains, accordingly, a curve 

 of the order 76, F». 



There are 76 conies lohich cut five straight lines, touch tioo given 

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



The curve k'^ has as many donble points on a\ as there are 

 conies of which the planes pass through r/,, which cut a,, . . , . a^, 

 and which touch the planes y, and ^2- I" order to find this number 

 we remark in the first place that the conies through two points 

 A and B of «, intersecting rt, and touching ff, and </>,, form a 

 surface of the eighth order. For in each plane through öj there lie 

 four conies satisfying these conditions, and «, is not a component 

 part of any such a degenerate conic. Hence eight conies intersecting 

 (7, and a, and touching (f, and 'f., pass through A and B and the 

 line «1 is an eightfold straight line of the surface formed by the 

 conies through A intersecting a^ outside A, cutting a, and a„ and 

 touching y, and y,. This surface is of the sixteenth order as appears 

 from its intersection with a plane through o, . a, is therefore a 

 sixteenfold straight line of the surface consisting of the conies the 



