116 Mr Richmond, On the parametric representation, etc. 



On the parametric representation of the coordinates of points 

 on a cubic surface in space of four dimensions. By H. W. 

 Richmond, M.A., King's College. 



[Received and read 8 March 1909.] 



By a cubic surface in space of four dimensions is here under- 

 stood the locus represented by a homogeneous equation of the 

 third degree in five variables, and the problem considered is a 

 method of expressing the ratios of the five variables as algebraic 

 functions of three parameters. Should the cubic surface possess a 

 double point, projection with that point as vertex leads at once to 

 a solution. 



If however the surface has no double point, take any straight 

 line L which lies wholly on the surface ; six such lines pass through 

 each point and twenty-seven lie in every space of three dimensions, 

 at least three of the latter being real. Through L and any point 

 P of the surface can be drawn an *S^2. whose intersection with the 

 surface will consist of the Hue L and a conic, which must intersect 

 L in two points, Qi and Qg- Conversely, if any two points Qi and 

 Q2 be chosen upon L, the tangent S-^s at Qi and Q.2 have in common 

 an S^ which contains L and a conic passing through Qi ^'^d Q^. 

 Thus if points of the line L are determined by a parameter, 0, 

 then to any two values of it <^i and <^2 correspond two points Q^ 

 and Q^, and consequently a conic passing through Q^ and Q>^. 

 Another parameter -»|r will define each point of the conic, and thus 

 the coordinates of each point of the surface will be algebraic 

 functions of <^i, <^2 and ■^. 



In fact the equation 



u"x + 2uvy + v^z 4- 2wX, -f- 2vyu, -\-p = (1), 



in which X and yu, are quadratic functions and p a cubic function 

 of X, y, z, represents a quite general cubic surface on which the 

 line L, cc= y = z= lies. A point on L is determined by a 

 parameter ^ if v — u .(f); and the tangent Ss at the point is 



X + 2y^ + z(f)^ = 0. 



Thus on the S2 common to the tangent S-Js at points where ^ 

 has the values (f)i and (j).2, 



X -.y : z :: (fy^cf).^ : - -^ ((/)i + (f)^) : 1 ; 



and if in (1) we substitute x = z . (f)i<f).2, y = — \z {(f>i + (po) and then 

 write M01 — v = z.-^, the problem is solved. 



