108 
ON THE CUBIC CURVES INSCRIBED IN A 
[400 
viz. □, V are the discriminants of the two functions respectively, and 0, R are 
simultaneous invariants of the two functions, R being in fact the resultant. The 
corresponding invariants of the functions V (c) . (a, h, k, b\x, y) 3 , and ( j, l, f\x, y) 2 are 
obviously c 2 D, V, c© and cR. 
The values of S and T are obtained from the Tables 62 and 63 of my “ Third 
Memoir on Quantics,” Phil. Trans, vol. cxlvi. (1856), pp. 627—647, [144], by merely 
writing therein g=i = 0. It appears that they are in fact functions of c 2 D, V, c© 
and cR; viz. we have 
S = V 2 + c©, 
T — 8V 3 + c (4>R + 12 V©) + ÆL 
The invariants of the sextic (*]£«, 2/) 6 > if for a moment the coefficients of this sextic 
are taken to be (a, b, c, d, e, f g), that is, if the sextic be represented by 
(a, b, c, d, e, f g][x, y) 6 are the quadrinvariant (= ag — 65/ + lace — 10cZ 2 ), Table No. 31 
and Salmon’s A., p. 203( 1 ), the quartinvariant, No. 34, and Salmon’s R, p. 203, the 
sextinvariant No. 35, and Salmon’s C., p. 204, and the discriminant, which is a function 
of the tenth order = a 5 g 5 + &c. recently calculated for the general form, Salmon, 
pp. 205—207, say these invariants are Q 2 , Q 4 , Q 6 and Q 10 . These several invariants 
are functions of the above-mentioned expressions c 2 D, V, c© and cR ; whence, con 
versely, these quantities are functions of the four invariants Q 2t Q 4 , Q 6 , Q 10 ; and the 
invariants S, T of the cubic curve, being functions of c 2 D, V, c© and cR, are also, 
as they should be, functions of the invariants Q 2 , Q it Q 6 and Q 10 of the sextic pencil 
(*$>> yY- 
To effect the calculation of Q.,, Q 4 and Q 6 , I remark that inasmuch as by a linear 
transformation, the quadric (j, l, /$#, yY may be reduced to the form 2Ixy, and that 
the invariants of (a, h, k, b\x, y) 3 and 2Ixy are 
□ = a 2 5 2 — 6abhk + 4 ak 3 + 4 bh 3 — 3 h-k 2 , 
V =-l\ 
© = — l (ab — hk), 
R = — 8 l 3 ab, 
hence, writing j = 0, f = 0, and writing also c=l, we may consider the sextic 
[(a, h, k, b\x, y) 3 f + 32l 3 a?y 3 , 
that is 
(a 2 , ah, i(2ak + 3h 2 ), ^(ab + 9hk + 161 s ), ±(2bh + 3k 3 ), bk, b 3 \x, yf, 
the invariants whereof are found to be functions of the last mentioned values of 
□ , V, ©, R- to pass to the given sextic (*$«, yY, put equal to 
c [(a, h, k, bjcc, y) 3 f + 4 [(j, l, f\x, yf] 3 , 
we have only to consider □, V, ©, R as having their before-mentioned general values, 
and to restore the coefficient c by the principle of homogeneity. 
1 The pages refer to Salmon’s Lessons Introductory to the Modern Higher Algebra (Second Edition, 1866). 
In the Fourth Edition, 1885, the values are given, pp. 260—265.