1895.] Cubic Surfaces containing 27 real straight lines. 9 



When looking at the diagram we see that a parabola {e.g. that 

 through D, a, A) being visible on the face ABC passes through 

 the point A, and being now on the further side of the solid branch 

 at A passes out of sight. On each solid branch there are evidently 

 three edges of the tetrahedron and three parabolas symmetrically 

 placed, and if we cut any solid branch transversally by a plane at 

 right angles to the line passing through its vertex and the centre of 

 the tetrahedron we obtain a symmetrical cubic oval, a diagram 

 of which is given, having its points of maximum and minimum 

 curvature at M and m, where the edges and parabolas respectively 

 meet it. There are also three infinite branches caused by the 

 section of the plane and the three other solid portions. 



Each edge of the tetrahedron coinciding with four straight 

 lines on the surface and three being at infinity there are still of 

 course twenty-seven straight lines on the surface. 



Some of the simpler cases of binodes yet remain to be con- 

 sidered that have equations of the form 



a/3y = KS/jlv. 



Construction of the surface. 



Take afty = K8fj,v as the equation of a cubic surface where 

 a , & y, $, p, v are of the first degree and K constant. Nine of its 

 twenty-seven straight lines may be at once obtained as the inter- 

 sections of the planes a, /3, y with 8, //., v respectively, and if we 

 take them to be represented by the numbers 4, 2, 12, 13, 14, 

 15, 9, 8, 7 the planes and their intersections are shewn by the 

 scheme 



If we take the lines forming a diagonal of this table, e.g. 4, 14, 

 7, we can find three straight lines on the surface that meet all of 

 these, and no more. 



By interchanging rows and columns we get in all six such 

 diagonals. Therefore there are 18 beside the 9 at first found, 

 making a total of 27. 



We can take it that 



5, 16, 17 meet 4, 14, 7. 6, 26, 27 meet 4, 8, 15. 



10, 18, 19 „ 13, 8, 12. 3, 22, 23 „ 13, 7, 2. 



1, 20, 21 „ 9, 2, 15. 11, 24, 25 „ 9, 12, 14. 



