290 ME. A. CAYLEY’S SEVENTH MEMOIE ON QUANTICS. 
247. It will be noticed how Tables 74 and 75 form a system involving only the 
qualities in (a, j3) contained in the first part of Table 73, and how, in like manner. 
Tables 76 and 77 form a system involving only the quantics in (a, /3) contained in the 
second part of Table 7 3 ; and, moreover, how in each pair of Tables the covariants, &c. 
correspond to each other as follows, viz. — 
R, S, T, 1 , H, C, D, P, Q, Y, Z to 
S, R, T, Y, P,Z, Q, H, D, 1,C. 
Thus in Table 74, — the formula for PI(ceU d-6/3HU), 
and in Table 75, — the formula for P (2aYU — 2j3ZU ), 
each of them involve the same factor 
J d,(l, 0 , -24S,..Ie.,/3)hU 
0. -24S,..I«,/3)-.HU, 
and so in all the other cases. 
248. The quantics in (a, j3) in each part of the foregoing Table 73 are covariantively 
connected together. In fact, considering the function (1, 0, — 24S, ..^a. j 8 )^ which 
for shortness I call G, we have 
G= (1,0, -248, /3)h 
IG= 0, 
JG = 4(648^- P)=4R, 
□ G=:: (IG)^ -27(JG)^=-432Il% 
HG=-4(S,T, .. 5;s(3)‘, 
<I>G= 2(T, 96S’, .. (3)'. 
The last-mentioned formulae, by the aid of Table 72, give rise to the following more 
general system in which they are themselves included. 
Table No. 78. 
xG+ 6 ,-a,HG = xG+ 6 /-xHG, 
H(AG+6i7.HG)= 36//,^G+A^HG, 
^(AG+6^7.HG)=(>d-216R(7,=’)2(T, 968^ (5f, 
I (xG-j-6(7jHG)= 
J (xG-f6/xHG)=4Pt(X^+216Ilf<,*), 
□ (aG+ 6 pTIG} = - 43214^ (X^- 216R/.o^)^ 
The expression for H(xGfi- 6 («/HG), putting therein X=0, shows that, to a numerical 
factor H.HG is equal to G, and hence, disregarding numerical factors, we may say 
that each of the quartics (1, 0, 248, (3)% ( 8 , T, ..%a, (3)*, is the Hessian of the 
other of them, and that the sextic (T, 968^, . . ^ 05 , /3)® is the cubicovariant of each of 
them. 
