9 
693] 
339 
693. 
A TENTH MEMOIR ON QUANTICS. 
[From the Philosophical Transactions of the Royal Society of London, vol. clxix., Part II. 
(1878), pp. 603—661. Received June 12,—Read June 20, 1878.] 
The present Memoir, which relates to the binary quintic (*\x, yf, has been in 
hand for a considerable time: the chief subject-matter was intended to be the theory 
of a canonical form which was discovered by myself and is briefly noticed in Salmon’s 
Higher Algebra, 3rd Ed. (1876), pp. 217, 218; writing a, b, c, d, e, f g,..,u, v, w to 
denote the 23 covariants of the quintic, then a, b, c, d, f are connected by the relation 
/ 2 = — a s d + a?bc — 4c 3 ; 
and the form contains these covariants thus connected together, and also e ; it, in fact, is 
(1, 0, c, f a-b - 3c 2 , a 2 e - 2cf\x, yf. 
But the whole plan of the Memoir was changed by Sylvester’s discovery of what 
I term the Numerical Generating Function (N.G.F.) of the covariants of the quintic, 
and my own subsequent establishment of the Real Generating Function (R.G.F.) of 
the same covariants. The effect of this was to enable me to establish for any given 
degree in the coefficients and order in the variables, or as it is convenient to express 
it, for any given deg-order whatever, a selected system of powers and products of the 
covariants, say a system of “ segregates ” : these are asyzygetic, that is, not connected 
together by any linear equation with numerical coefficients ; and they are also such 
that every other combination of covariants of the same deg-order, say every “congregate” 
of the same deg-order, can be expressed (and that, obviously, in one way only) as a 
linear function, with numerical coefficients, of the segregates of that deg-order. The 
number of congregates of a given deg-order is precisely equal to the number of the 
independent syzygies of the same deg-order, so that these syzygies give in effect the 
congregates in terms of the segregates: and the proper form in which to exhibit the 
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