## Invariant Sentence Examples

- 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. - The
**invariant**theory then existing was classified by them as appertaining to " finite continuous groups." - This expression of R shows that, as will afterwards appear, the resultant is a simultaneous
**invariant**of the two forms. - 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... - 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. - 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. - We cannot, however, say that it is an
**invariant**unless it is expressible in terms of the real coefficients. - 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. - 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. - (ab)i(ac)j(bc)k..., that the symbolic product (ab)i(ac)j(bc)k..., possesses the
**invariant**property. - If the forms be all linear and different, the function is an
**invariant**, viz. - 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. - 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. - - 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. - 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. - The second and third are those upon the solution of which the theory of the
**invariant**may be said to depend. - 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**. - If 0 be the degree of an
**invariant**j - aj aj a; oj =a ° a a o +al aa l +... - The Aronhold process, given by the operation a as between any two of the forms, causes such an
**invariant**to vanish. - 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
of the form, and fixing upon n-2**invariants**every other**invariants****invariant**is a rational function of its members. - The vanishing of this
**invariant**is the condition for equal roots. - 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. - X x x To form an
**invariant**or covariant we have merely to form a product of factors of two kinds, viz. - 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. - 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**. - 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. - 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 know that this x2 is an
**invariant**; i.e. - 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. - 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). - 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.