ME. A. CAYLEY’S SEVENTH MEMOIE ON QUANTICS. 
281 
and it did not occur to me how a similar covariant, such as M. Arojsthold’s which 
involves the two functions symmetrically, was to be formed. Let (A, B, C) be the first 
derived functions, («, b, c, f, g, h) the second derived functions of the cubic U, and 
(A', B', C') the first derived functions, («', b\ c\ f\ g\ h') the second derived functions of 
the Hessian HU, then disregarding numerical factors, we have 
0U =(bG -f , . . gh -af , . . XA', B', C'f, 
©^U=(5V-/X . . g'h'-a'f, . . XA , B, C 
and 
■4,={bc'+b’c-2ff,..ffh'+g'h-i,f-a'f, . . XA, B, CIA', B', C'); 
and considering U=0 as the equation of a curve of the third order, the equations 
0U = O, 0^U=O, %^ = 0 have the following significations, viz. 0U = O is the locus of a 
point, such that its second or line polar with respect to the Hessian touches its first or 
conic polar with respect to the cubic : 0^U is the locus of a point such that its second or 
line polar with respect to the cubic touches its first or conic polar with respect to the 
Hessian: and ^^ = 0 is the locus of a point such that its second or line polar with 
respect to the cubic, and its second or line polar with respect to the Hessian are reci- 
procals (that is, each passes through the pole of the other of them) with respect to the 
conic which is the envelope of a line cutting the first or conic polar of the point with 
respect to the cubic, and the first or conic polar of the point with respect to the Hessian 
in two pairs of points which are harmonically related to each other : such being in fact 
the immediate interpretation of the analytical formula. But this in passing. 
238. The formula (Tables 08 and 69 of my Third Memoir) for the disci iminants of 
the intermediates aU-f-6/3HU and OaPU+iSQU respectively are 
E( aU -f6/3HU)=[(l, 0, -24S, - 8T , ~48S^ 
K(6aPU+ /3QU)=[(48S, 8T, -96S^ -24TS, (3yjR. 
In M. Hermite’s paper in the ‘ Comptes Rendus,’ already referred to, there are given 
between these quantics in (a, (3) certain relations which (although less simple than the 
relations that will afterwards be obtained) I now proceed to investigate. Putting in 
the first formula ci-^(o=p, and in the second formula a -^(3 = 0, we have 
R(2>U -f6HU)=0, if (1, 0, -24S, - 8T, -48S^ J2^,iy=0, 
R(6^rpU-f QU)=0, if (48S, 8T, -96S^ -24TS, -P-16S^X^. 
which equations in^i, d, are about to be considered in place of the quantics from which 
they respectively arise. 
239. It is convenient to write* 
A=4S, 
B=^/P-64S" 
* A is (Aronholu’s aud) HERiiixE’s S, B is Hermixe’s Sj, and j), q, fl, A are Hermite’s S, A, d: 
there is a slight inaccuracy in three of his formulae, which shoidd be 
corresponding to formulae in the present memoir. 
MDCCCLXI. 2 Q 
