From the invariant a2 -2a 1 a 3 -2aoa4 of the quartic the diminishing process yields ai-2a 0 a 21 the leading coefficient of the Hessian of the cubic, and the increasing process leads to a3 -2a 2 a 4 +2a i a 5 which only requires the additional term-2aoa 6 to become a seminvariant of the sextic. A more important advantage, springing from the new form of S2, arises from the fact that if x"-aix n- +a2x n-2.
This arose from the study by Felix Klein and Sophus Lie of a new theory of groups of substitutions; it was shown that there exists an invariant theory connected with every group of linear substitutions.
F(a ' a ' a, ...a) =r A F(ao, a1, a2,ï¿½ï¿½ï¿½an), 0 1 2 n the function F(ao, al, a2,...an) is then said to be an invariant of the quantic gud linear transformation.
This expression of R shows that, as will afterwards appear, the resultant is a simultaneous invariant of the two forms.
Remark.-The invariant C is a numerical multiple of the resultant of the covariants i and j, and if C = o, p is the common factor of i and j.
Again, for the cubic, we can find A3(z) - -a6z6 1 -az 3.1 -a 2 z 2.1 -a 3 z 3.1 -a4 where the ground forms are indicated by the denominator factors, viz.: these are the cubic itself of degree order I, 3; the Hessian of degree order 2, 2; the cubi-covariant G of degree order 3, 3, and the quartic invariant of degree order 4, o.
1 A2B' Where The Denominator Factors Indicate The Forms Themselves, Their Jacobian, The Invariant Of The Quadratic And Their Resultant; Connected, As Shown By The Numerator, By A Syzygy Of Degreesorder (2, 2; 2).
Respectively: then the result arrived at above from the logarithmic expansion may be written (n)a(n x) = (n)x, exhibiting (n) $ as an invariant of the transformation given by the expressions of X1, X2, X3...
Such a symbolic product, if its does not vanish identically, denotes an invariant or a covariant, according as factors az, bz, cz,...
If ai, bx, cx be different forms we obtain, after development of the squared determinant and conversion to the real form (employing single and double dashes to distinguish the real coefficients of bx and cz), a(b'c"+b"c'-2 f'f") +b(c'a"+c"a'-2g'g") +c(a' +a"b'-2h'h")+2f(g'h"+g"h'-a' + 2g (h ' f"+h"f'-b'g"-b"g')+2h(f'g"+f"g'-c'h"-c"h'); a simultaneous invariant of the three forms, and now suppressing the dashes we obtain 6 (abc+2fgh -af t - bg 2 -ch2), the expression in brackets being the S well-known invariant of az, the vanishing of which expresses the condition that the form may break up into two linear factors, or, geometrically, that the conic may represent two right lines.
2 cos w xy+y 2 = X 2 +2 cos w'XY+Y2, from which it appears that the Boolian invariants of axe+2bxy-y2 are nothing more than the full invariants of the simultaneous quadratics ax2+2bxy+y2, x 2 +2 cos coxy+y2, the word invariant including here covariant.
The invariant theory then existing was classified by them as appertaining to " finite continuous groups."
By the x process of Aronhold we can form the invariant S for the cubic ay+XH:, and then the coefficient of X is the second invariant T.
Then of course (AB) = (ab) the fundamental fact which appertains to the theory of the general linear substitution; now here we have additional and equally fundamental facts; for since A i = Xa i +,ia2, A2= - ï¿½ay + X a2, AA =A?-}-A2= (X2 +M 2)(a i+ a z) =aa; A B =AjBi+A2B2= (X2 +, U2)(albi+a2b2) =ab; (XA) = X i A2 - X2 Ai = (Ax i + /-Lx2) (- /-jai + Xa2) - (- / J.x i '+' Axe) (X a i +%Ga^2) = (X2 +, u 2) (x a - = showing that, in the present theory, a a, a b, and (xa) possess the invariant property.
A property independent of the particular axes of co-ordinates used in the representation of the curve by its equation; for instance, the curve may have a node, and in order to this, a relation, say A = o, must exist between the coefficients of the equation; supposing the axes of co-ordinates altered, so that the equation becomes u' = o, and writing A' = o for the relation between the new coefficients, then the relations A = o, A' = o, as two different expressions of the same geometrical property, must each of them imply the other; this can only be the case when A, A' are functions differing only by a constant factor, or say, when A is an invariant of u.
A binary form of order n contains n independent constants, three of which by linear transformation can be given determinate values; the remaining n-3 coefficients, together with the determinant of transformation, give us n -2 parameters, and in consequence one relation must exist between any n - I invariants of the form, and fixing upon n-2 invariants every other invariant is a rational function of its members.
It may denote a simultaneous orthogonal invariant of forms of orders n i, n2, n3,...; degree 0 of the covariant in the coefficients.
Then if j, J be the original and transformed forms of an invariant J= (a1)wj, w being the weight of the invariant.
The fourth shows that every term of the invariant is of the same weight.
Moreover, if we add the first to the fourth we obtain aj 2w ak = 7 1=6, j, =0j, where 0 is the degree of the invariant; this shows, as we have before observed, that for an invariant w= - n0.
If 0 be the degree of an invariant j - aj aj a; oj =a ° a a o +al aa l +...
X x x To form an invariant or covariant we have merely to form a product of factors of two kinds, viz.
The linear transformation replaces points on lines through the origin by corresponding points on projectively corresponding lines through the origin; it therefore replaces a pencil of lines by another pencil, which corresponds projectively, and harmonic and other properties of pencils which are unaltered by linear transformation we may expect to find indicated in the invariant system.
If now the nti c denote a given pencil of lines, an invariant is the criterion of the pencil possessing some particular property which is independent alike of the axes and of the multiples, and a covariant expresses that the pencil of lines which it denotes is a fixed pencil whatever be the axes or the multiples.
Siace E2 + if + ~1, or ef, is necessarily an absolute invariant for all transformations of the (rectangular) co-ordinate axes, we infer that XE + un + v~ is also an absolute invariant.
From these formulae we derive two important relations, dp4 = or the function F, on the right which multiplies r, is said to be a simultaneous invariant or covariant of the system of quantics.
In addition, and transform each pair to a new pair by substitutions, having the same coefficients a ll, a12, a 21, a 22 and arrive at functions of the original coefficients and variables (of one or more quantics) which possess the abovedefinied invariant property.
In either case (AB) =A 1 B 2 -A 2 B 1 = (A/2)(ab); and, from the definition, (ab) possesses the invariant property.
Since (ab) = a l b 2 -a 2 b l, that this may be the case each form must be linear; and if the forms be different (ab) is an invariant (simultaneous) of the two forms, its real expression being aob l -a l b 0.
(ab)i(ac)j(bc)k..., that the symbolic product (ab)i(ac)j(bc)k..., possesses the invariant property.
May be always viewed as a simultaneous invariant of a number of different linear forms a x, x, c x, ....
May be a simultaneous invariant of a number of different forms az', bx 2, cx 3, ..., where n1, n 2, n3, ...
=n3, 'If' the forms a:, b:, cy 7 ...be identical the symbols are alternative, and provided that the form does not vanish it denotes an invariant of the single form ay.
The linear invariant a s is such that, when equated to zero, it determines the lines ax as harmonically conjugate to the lines xx; or, in other words, it is the condition that may denote lines at right angles.
If the forms be all linear and different, the function is an invariant, viz.
Possess the invariant property, and we may write (AB) i (AC)'(BC) k ...A P E B C...
- We have seen that (ab) is a simultaneous invariant of the two different linear forms a x, bx, and we observe that (ab) is equivalent to where f =a x, 4)=b.
The vanishing of this invariant is the condition for equal roots.
This expression gives the work done by a given wrench when the body receives a given infinitely small twist; it must of course be an absolute invariant for all transformations of rectangular axe~.
We cannot, however, say that it is an invariant unless it is expressible in terms of the real coefficients.
When either of the forms is of an order higher than the first (ab), as not being expressible in terms of the actual coefficients of the forms, is not an invariant and has no significance.
Possesses the invariant property.
It is always an invariant or covariant appertaining to a number of different linear forms, and as before it may vanish if two such linear forms be identical.
Moreover, its operation upon any invariant form produces an invariant form.
The second and third are those upon the solution of which the theory of the invariant may be said to depend.
We know that this x2 is an invariant; i.e.
The Aronhold process, given by the operation a as between any two of the forms, causes such an invariant to vanish.
In particular, when the product denotes an invariant we may transform each of the symbols a, b,...to x in succession, and take the sum of the resultant products; we thus obtain a covariant which is called the first evectant of the original invariant.
When the latter invariant, but not the former, vanishes, the displacement is equivalent to a pure rotation.
When the latter invariant, but not the former, vanishes, the system reduces to a single force.