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 ALGEBRAIC writing +d.2, -c^ for x2 and cc2, and noting that gx then becomes (gd), the above-written identity becomes {ad)(lc) + {bd){ca) + (cd){ab) — 0. . (II.) Similarly in (L), writing for Ci, c2, the cogredient pair -y2, +yl, we obtain (III.) axby-aybx = {ab){xy). Again in (I.) transposing ax(bc) to the other side and squaring, we obtain 2(ac)(6c)aa;6x=(6c)2a^ + (ac)2Ja.-(a&)24. (IV.) and herein writing d2, - dx for Xj, x2, 2(ac)(bc)(ad){bd) = (bc)ad)'i + (acfibd)2 - (abf^cd)2. (V.) 2 As an illustration multiply (IV.) throughout by a” cJ so that each term may denote a —covariant of an ?i*°. O/ /r x 17bxTl 1 c'xYl 1 2(ac)(bc)a a c71 ,, ,2 71,71-2 71 — 2, , a 71-2,7!. = (fic) cixbx cx +{ac) bxcx71-2 (/„i„7l-2,7l-2 b) dx x‘ x Each term on the right-hand side may be shown by permutation of a, b, c to be the symbolical representation of the same covariant; they are equivalent symbolic products, and we may accordingly write /i x71-1,71-1, , ,2ax71-2,71-2 2(ac)(bc)a bx c77-2 =[ab) bx cx71, x

293

FORMS

Similarly

„2 77-22,. ti-2 /:=«* 2*/f", ^2 /y = C/^i+W the second polar ; and in general the

A ti-22 fay 77-22 fa' polar is

fy=C/a +/22/2),i= j j % -2< {n-n)V w-a_ ay Vf) rs’dx^'dx a~, 1u.!' — a ?i! In symbolic form we may write f „ _{n-fi)/ 1 a lh2JLYf= fy n ^ dxl 0aJ2/ J n All the polars may be generated from ax by writing therein x1 + y1, x2+y2 for xv xx, respectively, for ax becomes (ax + ay)n and , 2 71 -2 ciy2, ... +,annan (ax+ayyi=ax+(i)ax Vy-f(2)^ y =fy + (1 )x /jf + (2 )x2/| + • • • + ^JyEx. gr. The first polar of a%=axay = (eqaq + a2x,1){alyl + a2y2) = alx1yi + a1a2(x1y2 + x2yi) + alfay^ a relation which shows that the form on the left is the product of the two covariants = ayxly1 + a1{x1y.2+x^yj + a^.2y2 , ,,2a71-2,71-2 {ab) bx and, c71x. x = («„*! + alx2)y1 + Kaq + a^2)y2 The identities are, in particular, of service in reducing symbolic ={y^+y^dx^) (fax*+2aifax2+a^i )• products to standard forms. A symbolical expression may be always so transformed that the power of any determinant factor Similarly the second polar of the binary cubic is {ab) is even. For we may in any product interchange a and b without altering its signification ((TqXj + a-lx2)yl + 2(a1aq + a^x2)y1y2 + (a^i + azfa)V 1 2m+1 2m+1 {ab) <p1= - {ab) <pz, which is axa?y or where becomes <p2 ^7 the interchange, and hence ^ d 3 2m 2m+1 l di + (a6) +Vi = |(«&) (0i - 02) i dxi ' ‘ dx2l) <4The operation of taking the polar results in a symbolic product and identity (I.) will always result in transforming fa - fa so as to make it divisible by {ab). and the repetition of the process, in regard to new cogredient sets of variables, must result in symbolic forms. It is therefore an Ex. gr. {ab){ac)bxcx= - {ab){bc)axcx invariant process, and all the forms obtained are invariants in regard to linear transformations, in accordance with the same = ^(«&)c*{(«c)&* - tf>c)ax} =^(a&)2c| ; scheme of substitutions, of the several sets of variables. An important associated operation is so that the covariant of the quadratic on the left is half the pro82 82 duct of the quadratic itself and its only invariant. To obtain the corresponding theorem concerning the general form of even 0aq0y2 caqB?/i ’ order we multiply throughout by {ab) cx and obtain which performed upon any polar causes it to vanish ; for 2m 2m - 1 U, — 1 . 32 n77 - /x /a = {n- y)ya a a rfm-1(aC)6xc2w-1-J(«5) fa ’ Y 2 x 0^07/5 i x Paying attention merely to the determinant factors there is no and conversely it can be shown that every function which it form with one factor since {ab) vanishes identically. ^ For two causes to vanish is a polar. factors the standard form is {ab'f ; for three factors {abf{ac) ; for It is usual to write four2 factors2 {ab'f and {abf{cdf; for2 five2 factors (abf{ac) and n m,n 1 / 02 02 777,77 = Ua {ab) {ac){de) ; for six factors {abf, («6) (&c) (ca)2, and {abf{cdf{eff. x by in.vcxfyy2 cxpyiJ ’x °y It will be a useful exercise for the reader to interpret the corresponding covariants of the general quantic, to show that some we have the theorem that G, performed upon any invariant of them are simple powers or products of other covariants of and form, produces an invariant form. lower degrees and order. Ex. gr. The Polar Process.—We are now going to introduce other sets „ 777.77 1, , ,  777— 1,77 — 1 of cogredient variables into the symbolic products. The yth polar £lax bx — {mnalb2 iiiHa2bf)ax ox , v of the binary form a x = / with regard to y is expressed by, ,, 777-1,77-1 — {ab)a by ,_ x x + 11= {al)y =fy = («A + fa T ~ faV* ' X Q?ax bx = {abfax by ; i.e., y. of the symbolic factors of the form are replaced by y others and in general in which yY, y2, replace xx, x2. By giving y the values 0, 1, 2, ...n we obtain in all n + 1 y-polars 12 cty> by — (ab) a^> by • in regard to x. They may be obtained by partial differential These are invariants of forms which are oinary in two sets of operations upon the form. Write in symbolic form variables and, by putting y = x, we obtain simultaneous inay nl variants of two binary forms in a single set of variables. {n - y.)r ^2 Observe the easy passage from a bipartite form to two unipartite Cx^dx forms. The polar of a product ax. bx is obtained by, first of all, so that n n writing a™.bx in the symbolic form ■A,$L= fx, dxx T31j} —a777.b, 77 —p777 + 77 > and let x x x fy=Av +Uji ; tPc m+n-k k. then ^ y~Px Py ’ 71—1 y, 71 — 1 n /. f~nax ax, n-1 nf,=nax c now, inp™+n =a™.bx, write x + y for x, so that a a f — Jy — x y’ (Px + xPy)m+ n = (ax + ^faY- (fix + X&2/)"’ ’ the first polar.
 * therefore
 * thus if