438 Mr. A. Cayley on the Nodal Curve of the 



or, as these may also be written, 



a q d-3abc + 2b 3 =0, 



a*e + 2abd-9ac? + 66 2 c=0, 

 abe-3acd + 2b*d =0, 



ad? -bH =0, 



ade—3bce + 2b(P =0, 



ae* + 2bde-9c*e + 6cd*=0, 

 be? -3cde + 2d 3 =0; 



which curve is in fact an excubo-quartic, — viz. a quartic curve 

 the partial intersection of a quadric surface, and a cubic surface, 

 having in common two non-intersecting right lines. To show 

 that this is so, I remark that the coefficients a, b, c, d } e f qua 

 linear functions of the four coordinates, satisfy a linear equation 

 which may be taken to be 



a + b + c + d+e = 0. 



This being so, the first form shows that the curve in question 

 lies on the quadric surface 



ac-b? + ?; (ad-bc)+ \(ae + 2bd-3c*) + ] i be-cd) + ce-d 2 =0. 

 or, as this equation may also be written, substituting for c, 



C ( a -l b ~ l 2 C -l d+e ) 

 -b 2 +^ad+^{ae + 2bd) + ^be-d*=0. 

 Substituting for c its value, this equation is 

 -(a + e + b + d)^(a + e)-b* + ^ad+±{ae + 2bd)+^be-d*=0, 



or, what is the same thing, 



9{a + e + b + d){a + e)+6(b* + d2)-S(ad+be)-(ae + 2bd) = 0. 

 Whence, finally, the equation of the quadric surface is 

 9a*+l7ae + 9e* 

 + 66 2 - 2bd+6d* 

 + 9ab + 9de + 6ad+6be=0; 

 and the curve lies also on the cubic surface 

 ad 2 -b?e=0. 



It only remains to show that these surfaces have in common 

 two right lines, and to find the equations of these lines. 



The cubic surface is a skew surface or " scroll " such that the 



