132 Professor Baker, Ninth point of two plane cubics 



and V, and on Q, and so projects from D into a point common to 

 the two plane cubics. This justifies the statement. 



Incidentally any two cubic curves in a plane are shown to be 

 the projections of two quartic curves in space lying on the same 

 quadric; and the plane problem is put in connexion with the space 1 

 problem of finding the remaining eighth intersection of three 1 

 quadrics with seven common points. 



To construct a twisted cubic curve with five given points 

 D, A, Pj, ^1, Ri to have a given line BC as chord, we may for 

 instance first construct a quadric surface by the intersection of 

 corresponding planes of two homographic axial pencils with DA, 

 BC as axes, three pairs of corresponding planes being those con- 

 taining Pj, Qi, Ri, and then construct a quadric surface by the 

 intersection of corresponding planes of two homographic axial . 

 pencils with DP^^, BC as axes, three pairs of corresponding planes 

 being those containing A, Qj^, Rj^. These quadric surfaces intersect 

 in the cubic curve required. 



It is seen that analytically each step requires only the solution 

 x)f linear equations. Indeed, if the conic through A, B, C, M, N 

 be written (referred to ^, P, C, D) as Ayz + Bzx + Cxy = 0, the 

 line 31N being x + y -\- z = 0, we may take for Q, the quadric 

 t {x + y + z) = Ayz + Bzx + Cxy. The general plane cubic curve 

 through the five points A, B, C, M, N may be taken to be 



{Ayz + Bzx + Cxy) {Ix + my + nz) + {x + y + z) x [qy + rz) = 0, 



and two cubics through these and P, Q, R may be found by solving 

 for the ratios of I, ni, n, q, r in the three equations obtained by 

 substituting the coordinates of P, Q, R. Corresponding to two sets 

 of ratios l-^ : m^ : n^ : q^ : rj, and l^ '. m^ : % • % '■ ^2 ^^ chosen, 

 there are two quadric surfaces 



t {l-^x + nijy + fijz) + X {q^y + r-f^z) =-- 0, 



t {I2X + m^y + n^z) + x (q^y + r^z) = 0, 



which intersect in a cubic curve containing D, A, P^, Qj^, Pj and 

 having BC for chord. The combination of these with the equation 

 of O will lead to a linear equation for Oj, from which is found. 

 Or the solution may be stated, naturally enough, without reference 

 to three dimensions. 



