igoj] Henderson — A Memoir. 85 



degenerate into a pair of straight lines. Here the plane 

 intersects the surface in three intersecting- straight lines (a 

 degenerate curve of the third order having three double 

 points) and the points of intersection of the lines taken in 

 pairs are the points of contact of the plane with the surface. 

 Now through each of the three lines in the plane there may 

 be drawn, besides the given plane, four triple tangent planes. 

 For these twelve new planes give rise to twenty-four new 

 lines upon the surface, making up with the former three lines, 

 twenty-seven lines upon the surface. It follows that every 

 straight line on the surface is met by ten others. 



If all the twenty-seven lines intersect in pairs, there would 

 be 351 points of intersection. But since each line is met by 

 ten other lines, there remain 16 lines by which it is not met 



and therefore there are — = 216 pairs of lines that do 



not intersect. Consequently there are 135 points of inter- 

 section. 



Since these 135 points, by threes, determined the triple tan- 

 gent planes, there are 45 triple tangent planes. 



Consider the three lines a, b, and c say, the complete inter- 

 section of the triple tangent plane it with the surface. Then 

 every other line / upon the surface must meet the triple tan- 

 gent plane in a point upon one {a say) of the three lines a, b, 

 and c, and accordingly must lie in a plane tri , passing through 

 a. Since the intersection of the surface by the plane rr i must 

 be a cubic curve, which is already composed of two straight 

 lines, the plane ir i meets the surface in a third straight line 

 /', and therefore must be a triple tangent plane. Hence /' 

 must be one of the given 27 lines and it appears that there 

 can be but 27 lines upon a cubic surface. 



§4. Salmon'' s Notation for the Twenty- Seven Lines.* 



Lemma. The general equation of the cubic surface may be 



*Oamb. and Dublin Math. Journal (1849), Vol. IV., pp. 252-260. 



