Mr Grace, Note on the Rational space curve etc. 
27 
Note on the Rational space curve of the fourth order. By 
J. H. Grace, M.A., Peterhouse. 
[Read 26 November 1900.] 
1 . The relations between the geometry of points on a rational 
quartic and the algebraic theory of a binary quartic have been 
frequently discussed \ but the application of similar methods 
to the line-geometry of the tangents to the curve has not, as far 
as I know, been previously remarked. 
If we supposed the coordinates x, y, z, w of a point on the 
curve to be expressed as quartic functions of a simple para- 
meter \ so that 
*=/lW J/=/lW 2=/lW. W=f a (\), 
then the six coordinates of the tangent at any point are the six 
Jacobians of the f’s taken in pairs. For convenience I denote 
these by 
J 23 ) J 31 ? J 12 ) J 14 ) J 24 5 Ju- 
2. Each of the four f’s is apolar to one and the same 
quartic f and thus the condition that four points should be 
coplanar is that the quartic giving their parameters should be 
apolar to f 
In like manner each of the sextics J is apolar to one and 
the same sextic <£, and the condition that the tangents at six 
points should belong to the same linear complex is that the 
sextic giving their parameters should be apolar to <j>. 
3. Let us suppose that f= a* 4 , </> == c A 6 , and that a, 6, 7 , 8, e, f 
are the parameters of six points on the curve ; then it is well 
known, it can be easily verified, and it is indeed obvious from 
the simple properties of transvectants that a, /3, 7, 8 are co- 
planar if 
a a a p a y as = 0 . 
Further, the tangents at the six points belong to the same 
linear complex if 
CaCpCyCsC^ = 0 . 
Hence just as f=af gives the points at which a plane 
contains four consecutive points, so <j> = cf gives the points at 
which a linear complex contains six consecutive tangents to the 
curve. 
1 See Mr Richmond’s recent and comprehensive Memoir (Gavib. Phil. Soc. Trans. 
Yol. xix. Pt. 1 .), in which full references will be found. 
