( 5:{(; ) 



of contact with two conies out of (he pencil (c//, ) counting two 

 times. Likewise do the 24 points of intersection of the conic 



hi with c 12 consist of the five basepoints differing from A'i counting- 

 four times and the imaginary points of contact with the tangents 

 through A'i counting two times. 



3. From the investigations of F. Klein and H. G. Zeutiien dating 

 from 1873 and 1875 it lias become evident that the surface 5' with 

 27 real right lines has ten openings and the parabolic curve s ls ~has 

 ten oval branches. In connection with this we find : 



"The locus c 13 has ten oval branches." 



We ask which situation of the six basepoints A'i corresponds to 

 the particular case of the "surface of diagonals" of Clebsch, in which 

 the ten oval branches of the curve s" have contracted to isolated 

 points. In this case the fifteen lines with real asymptotic points, i.e. 

 in our case the lines c'ik, pass ten times three by three through a 

 point; this is satisfied by the six points consisting of the five vertices 

 of a regular pentagon and the centre of the circumscribed circle. 



Fig. 1. 



What is more, each six points having the indicated situation can be 

 brought, by central projection to this more regular shape. The ten 

 meeting-points of the triplets of lines then form the vertices of two 

 regular pentagons (fig. t). The curve c 12 corresponding to these six 

 basepoints then consists of merely isolated points, namely of fourfold 



