1895.] Cubic Surfaces containing 27 real straight lines. 7 



Three drawings from photographs of a model shew these 

 openings, the fourth being a view of the under part. [Plate I. 

 Figs, i — iv.] 



The lines on the solid upper portion which are the same as 

 those on the lower hollow portion form a double six, these lines 

 each pass through one opening. 



The lines through each opening form a double six. 



The lines between each opening and the next form a double 

 six. 



Twelve lines pass through two openings namely 17, 23, 13, 19, 

 1, 9, 7, 26, 11, 16, 4, 22. 



Three lines 8, 18, 27 do not pass through any opening. 



One conic node. 



Take the lines 1,13; 14, 6 ; 4, 11 ; 22, 16 ; 23, 17 ; 10, 2 and let 

 them move up to one another namely 1 to 13, 14 to 6, &c. so as 

 to form six lines only r lt r. 2) ..., r 6 . 



These lines will now all pass through a point, and the opening 

 through which they go will close up to this point and become a 

 conic node. We now have 15 mere lines and 6 rays of a conic 

 node. 



Two conic nodes. 



If the two conic node rays r 3 , i\ now coincide, another opening 

 closes up, namely that through which the doable six 4, 22, 11, 16, 

 19, 9, 25, 5, 7, 26, 21, 12 passes. r 3 , r 4 now coincide to form an 

 axis A x joining the two conic nodes. The line 27 becomes a 

 transversal meeting the axis. The lines 19, 9 ; 25, 5 ; 7, 26 ; 21, 

 12 coincide by pairs to form four rays of the second conic node 

 } \> r 9 , r 9 , r 10 . Six more lines remain, each of which meets the 

 transversal, namely 3, 8, 15, 18, 20, 24. 



Three conic nodes. 



Two rays from each conic node r x , r 5 ; r 7> r g may be made to 

 coincide. The third opening now becomes a point joined to the 

 other nodes by axes A,, A s . 3, 20; 15, 24 unite to form r n> r 12 

 rays of the third node. 8, 18, 27 are the transversals. 



Four conic nodes. 



The six conic node rays in this case may be made to coincide 

 by pairs. We then get a fourth conic node caused by the closing 

 up of the opening at the base of the model. 27, 8, 18 still remain 

 transversals. 



Conic nodes may also have been found by considering the solid 

 portions between two openings as diminishing in size, we then get 

 a conical point formed by the meeting of two solid conical portions, 

 instead of hollow ones. 



