object of the present memoir is the further development of the theory of binary ntics; it should therefore have preceded so much of my third memoir, t. 147 (1857), 27, as relates to ternary quadrics and cubics. The paragraphs are numbered conously with those of the former memoirs. The first three paragraphs, Nos. 62 to 64, te to quantics of the general form (*)(
x, y
,..)
m
, and they are intended to complete series of definitions and explanations given in Nos. 54 to 61 of my third memoir; 68 to 71, although introduced in reference to binary quantics, relate or may be dered as relating to quantics of the like general form. But with these exceptions memoir relates to binary quantics of any order whatever: viz. No. 65 to 80 relate he covariants and invariants of the degrees 2, 3 and 4; Nos. 81 and 82 (which are xluced somewhat parenthetically) contain the explanation of a process for the alation of the invariant called the Discriminant; Nos. 83 to 85 contain the definitions he Catalecticant, the Lambdaic and the Canonisant, which are functions occurring in ’essor Sylvester’s theory of the reduction of a binary quantic to its canonical form; Nos. 86 to 91 contain the definitions of certain co variants or other derivatives coned with Bezouts abbreviated method of elimination, due for the most part to Pro- Sylvester, and which are called Bezoutiants, Cobezoutiants, &c. I have not in present memoir in any wise considered the theories to which the catalecticant, &c. the last-mentioned other co variants and derivatives relate; the design is to point ind precisely define the different covariants or other derivatives which have hitherto ented themselves in theories relating to binary quantics, and so to complete, as far ay be, the explanation of the terminology of this part of the subject. If we consider a quantic (
a, b
,..)(
x, y
,...)
m
an adjoint linear form, the operative quantic (
a, b
,..)(∂
e
, ∂
n
,...)
m
ore generally the operative quantic obtained by replacing in any covariant of the quantic the facients (
x
,
y
,..) by the symbols of differentiation (∂
e
, ∂
n
,...) ore generally the operative quantic obtained by replacing in any covariant of the quantic the facients (
x, y
, ..) by the symbols of differentiation (∂
e
, ∂
n
,...) (which ative quantic is, so to speak, a contravariant operator), may be termed the
Pro
- r; and the Provector operating upon any contravariant gives rise to a contravariant, h may of course be an invariant. Any such contravariant, or rather such con-iriant considered as so generated, may be termed a
Provectant
; and in like manner operative quantic obtained by replacing in any contravariant of the given quantic the facients (
ξ
,
n
..) by the symbols of differentiation (∂
x
, ∂
y
,...) (which opera quantic is a covariant operator), is termed the
Contraprovector
; and the contraprove operating upon any covariant gives rise to a covariant, which may of course be an irriant. Any such covariant, or rather such covariant considered as so generated, may termed a
Contraprovectant
.
Mathematical solutions to the problem of creating basic decenter optical mirror models from the coma and astigmatism correction conditions
