Professor Baker, Ninth point of two plane cubics 131 



On the construction of the ninth point of intersection of two plane 

 cubic curves of which eight points are given. By Professor H. F. 

 Bakee. 



[Read 3 May 1920.] 



Cayley has collected, in a paper reprinted in Vol, iv of his 

 Papers, pp. 495-504 {Quart. J., v, 1862), the various solutions given 

 of this problem, regarded as a problem of plane geometry, by 

 Pliicker, Weddle, Chasles and Hart, depending for the most part 

 on the generation of a plane cubic curve (two points at a time) 

 by the intersection of a pencil of lines and a homographic pencil 

 of conies. So far as I have been able to notice, geometrical con- 

 ceptions present themselves to an unbiassed child in the first 

 instance as three dimensional, and he feels it to be an abstraction 

 to regard plane geometry as self-contained; the discussion of the 

 most natural Axioms of geometry seems also to point in this direc- 

 tion; and the most valuable part of a training in geometry would 

 seem to lie in the cultivation of a faculty for visualisation of 

 relations in space. However these things may be, it appears to 

 me always to be an interesting extension when a property of space 

 is shown to follow from a property in space of higher dimensions, 

 this being generally accompanied by the removal of some artifi- 

 ciality. Thus, I regard the very simple example which now follows 

 as being logically at least as fundamental as a proof in the plane. 



Let A, B, C, M, N and P, Q, R be the eight given coplanar 

 points. Take a point D outside the plane of these. There are qo ^ 

 quadric surfaces containing A, B, C and the lines DM, DN; let Q 

 be one of these (other than that consisting of the planes ABC, 

 DMN). Let DP, DQ, DR meet this quadric again in P^, Q^, Rj^. 

 A definite twisted cubic curve can be drawn through D, A, Pj, Q^, R^ 

 to have BC as a chord (see below). This cubic curve, meeting D. 

 in D, A, Pj, Qi, Pj, meets O in a further point, say 0^. If DO^ 

 meet the original plane in 0, this is the ninth point required. 



For the space cubic is the intersection of two quadric surfaces 

 drawn through D, A, P^, Q^, Pj, both having the line BC as a 

 generator; denote these by U and V. The quartic space curve of 

 intersection of U with O contains D, A, B, C, P^, Qi, R^, and 

 meets the generators DM, DN of Q; this curve then projects 

 from D on to the original plane into a cubic curve containing the 

 eight given points A, B, C, P, Q, R, M, N. The curve of inter- 

 section of V with Q. projects from D into another cubic through 

 these eight points. The point 0^, on the space cubic, lies on U 



9—2 



