( 535 ) 



points the 27 right lines of S\ supposed In be real, are to be 

 divided into two groups, into a group of 12 lines with imaginary 

 asymptotic points, the lines 



«, a 2 a z a 4 a 5 a s I 



b i K h * b A l h b e I 



of a doublesix and into a group of 15 lines c la , c lt , . . . , c tt with 

 real asymptotic points. If to the six basepoints A'; of the linear 

 system of the cubics the six lines a,- correspond - - and this case 

 we shall in the following continually have in view — then to the 



six lines bi correspond the six conies hi through all the 

 basepoints except A'i and to the fifteen lines ca correspond the 



connecting lines dik = (A'i, A' k), whilst to the systems of conies (a]) 



in planes through m , (bi) in planes through bi , (c ( x- ) in planes 

 through Cik correspond successively the pencils of the curves of the 

 linear system with A'i as doublepoint, the lines (b'i) through A'i 



and the conies (cik) through the four basepoints differing from 

 Ai and A]-. The situation of the six points A'i is then such that 

 each of the fifteen lines c'n is touched in real points by two conies 



of the pencils (Cik), whilst on the other hand the points of contact 



of the tangents out of the points A'i to the conies b; are imaginary 



so that each point A'i lies within the conic bi with the same index. 



2. As a matter of fact all real points of a line / of S' are hyper- 

 bolic points of this surface with the exception of the two asymptotic 

 points of this line showing a parabolic character ; whilst each of 

 these asymptotic points is point of contact of / with a conic lying 

 on *S 8 , / touches in both points the parabolic curve »S 18 . If we apply 

 this to each of the six lines a;, imaged in the points A'i, and it' 

 we consider that to a definite point P of a,- corresponds the point 

 P' lying infinitely close to A'i connected with A'i by a line of 

 detinite direction (Versl., vol. I, pag. 143) we imd immediately : 



"The six basepoints A'i of the linear system are fourfold points 

 of the curve c" of a particular character, consisting of the combi- 

 nation of two real cusps with conjugate imaginary cuspidal tangents, 

 the cuspidal tangents of the curves out of the system with 

 a cusp in A'i". 



The twelve points of intersection of the line c'n- with c 13 consist 

 of the isolated points A'i,A'k counting four times and the real points 



