I^IE. A. CAYLEY’S SEVENTH IHEMOIE ON QUANTICS. 291 
249. Similarly, if the function (48S, 8T, . . ^ 05 , (Sy is for shortness called G, then we 
have 
G =(48S, 8T, 
IG =0, 
JG =4(64S^-T7=4R^ 
□ G=(IG)^-27(JG)^=--432E^ 
HG = -4(P+192S^ . . %oc, jSy, 
OG = -2(-8P+4608TS^ . . J_a, /3)h 
The last-mentioned formulae, by the aid of the same Table 72, give rise to the more 
general system in which they are themselves included. 
Table No. 79. 
>.G-h6/!//HG = 
H(?.G+6(t4HG)= 
0(xG+6/u-HG)=: 
I (>.G+ 6|y.HG) = 
xG-j-dii^HG, 
36EyG+?v"HG, 
(X^-216Ey)x -2(-8P+4608TS^ . .^a, (Bf, 
72n^Xf^, 
J (XG+6(U;HG)= 4E^(X^+216Ey), 
□ (xG-l-6f^RG)=-432RXx^-216Ryy. 
The expression for H(>.G+6('>6HG), putting therein X=0, shows that, ta a numerical 
factor _pres, H . HG is equal to G ; so that, disregarding numerical factors, we may say 
that each of the quartics (48S, T, .. ^a, fB)*, (T^-}-192S®, . .'Xcc, (5)*, is the Hessian of 
the other of them, and that the sextic ( — 8T^+4608TS®, . . ^a, j3)® is the cubicovariant 
of each of them. 
250. But besides this, the quantics in (a, j3) in the two parts of the Table 73 are 
linearly connected together : the linear relations in question are in fact the equations 
whereon depend the expressions for the invariants in Tables 76 and 77 as deduced from 
those in Tables 7 4 and 7 5 ; and in the order of proof, they precede the formulse in these 
four Tables. The linear relations are 
Table No. 80. 
(1,0,-248,.. J-2T«-16S^/3, 8S«+ T/3)^=-16E (P+192S^ . . /3)‘, 
(S, T, . . ^-2Ta-16S^/3, 8S«+ T,3y= E^(48S, 8T, . . fB)\ 
(T, 96S^ . . ^-2T«-16S^/3, 8Sa+ Tj3)®=- 8Ir(-8P+4608TS^ . . (Bf. 
(488, 8T, . . ^ T«+168^|3, -88a-2Ti3)^=-16E^(8, T, . . (B)\ 
(P-f 1928^ . . % -88a-2T/3)^=- E^l, 0, -248, . . IB)\ 
(-8P+4608T8^ Ta-f 168^/3, -SSoc-2T(By=- 8E‘(T, 968^ . . %oc, ftf. 
