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 <j> 
is a co variant 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 
<p 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, + Qmx 2 y 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, /3, 7, 8, e, f belong to a linear 
complex if 
Ca,C(}CyCfrC e Cg = 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 a 5 c^ for all values of f, 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 f. 
Now taking f in the canonical form x 4, + 6mx 2 y 2 + y 4 , </> will be 
xy (a? — y 4 ) and the quartic has to be apolar to 
d 2 (f> d 2 (f) d 2 (f) 
dx 2 ’ dxdy ’ dy 2 ' 
It follows at once that the quartic is of the form 
a? + yx 2 y 2 + y 4 , 
i.e. of the form H + \f when H is the Hessian of f 
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. 
