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A FOURTH MEMOIR UPON QUANTICS. 
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77. I do not stop to consider the theory of the lineo-linear covariants of two 
quantics, but I derive the quadricovariants of a single quantic directly from the 
quadrinvariant. Imagine a quantic of any order even or odd. Its successive even 
emanants will be in regard to the facients of emanation quantics of an even order, 
and they will each of them have a quadrinvariant, which will be a quadricovariant of 
the given quantic. The emanants in question, beginning with the second emanant, are 
(in regard to the facients of the given quantic assumed to be of the order m) of the 
orders m — 2, m — 4,... down to 1 or 0, according as m is odd or even, or writing 
successively 2p + l and 2p in the place of m, and taking the emanants in a reverse order, 
the emanants for a quantic of any odd order 2p+l are of the orders 1, 3, 5... 2p — 1, 
and for a quantic of any even order 2p, they are of the orders 0, 2, 4 ... 2p — 2. The 
quadricovariants of a quantic of an odd order 2p + 1, are consequently of the orders 
2, 6, 10... 4p — 2, and the quadricovariants of a quantic of an even order 2p, are of 
the orders 0, 4, 8 ... 4p — 4. We might in each case carry the series one step further, 
and consider a quadricovariant of the order 4p + 2, or (as the case may be) 4p, which 
arises from the 0th emanant of the given quantic; such quadricovariant is, however, 
only the square of the given quantic. 
78. In the case of a quantic of an evenly even order (but in no other case) we 
have a quadricovariant of the same order with the quantic itself. We may in this 
case form the lineo-linear invariant of the quantic and the quadricovariant of the same 
order: such lineo-linear invariant is an invariant of the given quantic, and it is of 
the degree 3 in the coefficients, that is, it is a cubinvariant. This agrees with the 
before-mentioned theorem for the number of cubinvariants. 
79. In the case of the quartic (a, b, c, d, e§oc, y) 4 , the cubinvariant is, by the 
preceding mode of generation, obtained in the form 
e (ac — b 2 ) — 4df (ad — be) 4- 6c£ (ae — 4bd + 3c 2 ) — 46| (be — cd) + a (ce — d 2 ), 
which is in fact equal to 
3 (ace — ad 2 — b 2 e + 2 bed — c 3 ); 
and omitting the numerical factor 3, we have the cubinvariant of the quartic. 
80. In the case of a quantic of any order even or odd, the quadrinvariants of the 
quadricovariants are quartinvariants of the quantic. But these quartinvariants are not 
all of them independent, and there is no obvious method grounded on the preceding 
mode of generation for obtaining the number of the independent (asyzygetic) quartin 
variants, and thence the number of the irreducible quartinvariants of a quantic of a 
given order. 
81. I take the opportunity of giving some additional developments in relation to 
the discriminant of a quantic 
(a, b, ... b', y) m . 
To render the signification perfectly definite, it should be remarked that the discriminant 
contains the term a w_1 a m_1 , and that the coefficient of this term may be taken to be