28 Mr Grace, Note on the Rational space curve 



As the points given by </> possess a projective property with 

 reference to the curve, it follows from general principles that <£ 

 is a covariant of f, and being of degree six it must be the sextic 

 covariant. 



To verify that this is actually so, we need only prove that 

 </> is apolar to J 12 , say, i.e. we have to establish the following 

 algebraical theorem : — 



If two quartics are each apolar to a third quartic, then the 

 Jacobian of the former two is apolar to the sextic covariant of 

 the third. 



Taking the third in the canonical form x 4 + QmaPy 2 + y 4 this 

 is easy. 



In general terms then we may say that <£ bears the same 

 relation to the geometry of the tangents as f does to the geometry 

 of the points of the curve. 



4. I shall content myself with two deductions from the fore- 

 going principles. 



I. The tangents at a, ft, 7, 8, e, £ belong to a linear 

 complex if 



CaPfiCyClCfit = 0, 



and given five of the tangents this equation generally determines 

 the sixth uniquely. 



If, however, the five be such that any linear complex containing 

 four also contains the fifth, then the above equation must be 

 satisfied by all values of f ; hence the quintic giving a, /3, 7, 8, e 

 must be apolar to C\C$ for all values of £, i.e. it must be apolar 

 to all first polars of <£. 



In like manner the tangents at a, /3, 7, 8 are generators of 

 a hyperboloid if any linear complex containing three also contains 

 the fourth, and thus the above equation must be identical in 

 e and f. Hence the quartic giving a, /3, 7, 8 must be apolar to 

 C\%C£ for all values of e and £. 



Now taking f in the canonical form x i + 6mcc 1 y 2 + y 4 , <f> will be 

 xy (x 4, — y 4 ) and the quartic has to be apolar to 



8^ dty_ 8^ 

 dx 2 ' dxdy ' dy 2 ' 



It follows at once that the quartic is of the form 



x 4 + /xx 2 y' 2 + y 4 , 



i.e. of the form H +\f when H is the Hessian of/! 



Hence four tangents to the curve are generators of a hyper- 

 boloid when, and only when, the quartic giving their parameters 

 is of the form H + \f= 0. 



