[157 
157. 
ON THE TANGENTIAL OF A CUBIC. 
[From the Philosophical Transactions of the Royal Society of London, vol. xlviii. for the 
year 1858, pp. 461—463. Received February 11,—Read March 18, 1858.] 
In my “ Memoir on Curves of the Third Order ” (*), I had occasion to consider a 
derivative which may be termed the “tangential” of a cubic, viz. the tangent at 
the point (x, y, z) of the cubic curve (* \x, y, z) 3 = 0 meets the curve in a point 
(£, y, £)> which is the tangential of the first-mentioned point; and I showed that when 
the cubic is represented in the canonical form ot? + y 3 + z 3 + Qlxyz = 0, the coordinates of 
the tangential may be taken to be x (y 3 — z 3 ) : y (z 3 — x 3 ):z (x s — y 3 ). The method given for 
obtaining the tangential may be applied to the general form {a, b, c,f, g, h, i,j, k, Ifx, y, z) 3 : 
it seems desirable, in reference to the theory of cubic forms, to give the expression of 
the tangential for the general form 2 ; and this is what I propose to do, merely indicating 
the steps of the calculation, which was performed for me by Mr Creedy. 
The cubic form is 
(a, b, c, f, g, h, i, j, k, IJx, y, z) 3 , 
which means 
ax? + by 3 + cz 3 + 3fy 2 z + 3gz 2 x + 3 hx-y + 3 iyz 2 + Sjzx 2 + Shxy 2 + Qlxyz; 
and the expression for f is obtained from the equation 
^ = {b, f i, cf(j, f c, i, g, Ifx, y, z)\ -ih, b, i, f l, kfx, y, z) 2 ) 3 
~ («, b, c, f g, h, i, j, k, Ifx, y, z) 3 ((&x + 30), 
1 Philosophical Transactions, vol. cxlvii. (1857), [146]. 
2 At the time when the present paper was written, I was not aware of Mr Salmon’s theorem (Higher 
Plane Curves, p. 156), that the tangential of a point of the cubic is the intersection of the tangent of the 
cubic with the first or line polar of the point with respect to the Hessian; a theorem, which at the same 
time that it affords the easiest mode of calculation, renders the actual calculation of the coordinates of the 
tangential less important. Added 7th October, 1858.—A. C. 
558