84 Journal of the Mitchell Society. \_june 



since the equation in r must be identically satisfied, i. e., for 

 all values of r. Since in this case the equation of the surface 

 is of the third degree, there result four conditions. But the 

 equations of a straight line involve four disposable constants, 

 and, as the number of conditions to be fulfilled is exactly 

 equal to the number of disposable constants in the equations 

 of the straight line, it follows that every surface of the third 

 order must contain a finite number of straight lines, real or 

 imaginary, lying entirely upon it. 



§2. Number of Straight Lines upon a Cubic Surface. 



Suppose we pass a plane * through a point P outside the 

 surface and through a straight line / lying upon the surface. 

 Then 7r meets the surface in the line /, and a conic C besides 

 (since the curve of intersection is a degenerate cubic), i. e. 

 meets the surface in a section having two double-points and 

 therefore by definition is a double-tangent plane. These 

 double-tangent planes tr to the cubic surface are also double- 

 tangent planes to the tangent cone, vertex P. Now since to 

 every plane ir corresponds one straight line / lying entirely 

 on the surface and as there are twenty-seven* (« = 3) double- 

 tangent planes to the tangent cone, vertex P, therefore there 

 are twenty -seven straight lines /on the cubic surface.f 



§3. Triple Tangent Planes. 



By properly determining the plane passed through any 

 straight line / upon the cubic surface, the conic C (§2) will 



•Salmon, Geom. of Three Dimensions, 4th edition, §286 gives 



^-(w — l)(n — 2)(ns — n 2 + n — 12) 

 a 



as the number of double-tangent planes, drawn through a point P to a 

 surface of the nth degree. 



tFor other proofs compare R. Sturm, Flachen dritter Ordnung, Kap. 2, 

 §20, and Cayley, Ooll. Math. Papers, Vol. I., No. 76 (445-466). 



