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ALGEBRA I C

to which the quartic is in general reducible is, f=xix + Qmxl1xi2+x*2, involving one parameter m; then + x^) + 2(l - dm )x‘^, t = 2(l + 3m2) ; j = 6m(l - m)2 ; < = (l - 9m2)(x2 - x2)(a:2 + a;2^. The sextic covariant t is seen to be factorizable into three quadratic factors $$\phi = x_1, x_2, \psi =x_{1}^{2} + x_{2}^{2}, \psi = x^2_1 - x^2_2 $$, which are such that the three mutual second transvectants vanish identically ; they are for this reason termed conjugate quadratic factors. It is on a consideration of these factors of t that Cayley bases his solution of the quartic equation. For, since — — he compares the right - hand side with the cubic resolvent yfc3-i/, of /= 0, and notices that they become identical on substituting A for Ic, and —f for X ; hence, if Aq, Jc2, he the roots of the resolvent, -2^=(A + k1f)(A + Jc2f)(A + Jc3f); and now, if all the roots of/ be different, so also are those of the resolvent, since the latter, and /, have practically the same discriminant ; consequently each of the three factors, of - 2t2, must be perfect squares and taking the square root and it can be shown that 0, x> 'Z' are the three conjugate quadratic 2factors of t above mentioned. We have A + k1f=1, A + £2/=x, A + &3/= V/2, an(i Cayley shows that a root of the quartic can be expressed in the determinant form the remaining roots being obtained by vary1 > > Qy ing the signs which occur in the radicals 1 > > Xy 0), Xy, 'p})- The transformation to the normal form reduces the quartic to a quadratic. The y =0 new variables are the linear factors of <£. y2=l If is any one of the conjugate quadratic factors of t, so that, in determining rx, sx from /A + Aq/ = 0, Aq is any root of the resolvent. The transformation to the normal form, by the solution of a cubic and a quadratic, therefore, supplies a solution of the quartic. If (A/x) is the modulus of the transformation by which <4 is reduced to the normal form, i becomes (X/x)4i, and j, z i hence ^ is absolutely unaltered by transformation, and is termed ^ i3 2 (l+3m2)2 we the absolute invariant. Since therefore y = x9 mql ^ - sta have a cubic equation for determining m2 as a function of the absolute invariant. Remark.—Hermite ;has shown {Crelle, Bd. lii.) that the substitution, z = reduces x^::Xl ^}le form J f V/ dz 2i V 3 2 +i3 The Binary Quintic.—The complete system consists of 23 forms, of which the simplest are/= ax ; the Hessian H = (/, /')2 = {abyaxbx the quadratic covariant i=(f,f,')i:=(ab')‘laxbx ; and the nonic covariant T = (/, (/', /02)1 = (/, H)1 = («H)a4Hj = (ab)Xca)albl4; the remaining 19 are expressible as transvectants of compounds of these four. There are four invariants (i, i')2;2 4(i3, H)63;5 (f2,4i5)108; (ft, i7)u four linear forms (/, i ) ; (/, i 2) 4; (i, T)3 ;5 (i5, T)9 three quadratic forms i2 ; (H , 2i 3) ; (H, i) three cubic forms (/, i) ; (/, i ') ;2 3(i3, T)8 two quartic forms (H ,i)2; (H, i ' ) . three quintic forms /; (/, i)1; (i2, T)4 two sextic forms H ; 2(H, i)1 one septic form (i, T) one nonic form T. We will write the3 cubic covariant (f,i)2=j, and then remark that the result, (/,i) = 0, can be readily established. The form j is completely defined by the relation (/,i)3 = 0 as no other covariant possesses this property. Certain covariants of the quintic involve the same determinant factors as appeared in the system of the quartic ; these are /, H, i, T, and j, and are of special importance. Further, it is convenient to have before us two other quadratic covariants, viz.,

FORMS 2

r = (j,j) jxjx ", 0 = (ir)ixTx ; four other linear covariants, viz., a = - (jifjx ; (3 = (ia,ix; 7 = (ra)rx ; S = (rp)Tx. Further, in the case of invariants, we write A = (i,i’)2 and take three new forms B = (i,r)2; C = (t,t')2; K,= (/3y). Hermite expresses the quintic in a forme-type in which the constants are invariants and the variables linear covariants. If a, (3 be the linear forms, above defined, he raises the identity ax(af}) = af aft) - j3x(aa) to the fifth power (and in general to the power n) obtaining (a/3)y=(«/3)54 - 5(a(3)4(aa)alj3x+ ... - (aa)5/3|; and then expresses the coefficients, on the right, in terms of the fundamental invariants. On this principle the covariant j is expressible in the form R2; = 53 + |B52a + ^AC5a2 + |c(3AB - 4C)a3 when 5, a are the above defined linear forms. Hence, solving the cubic, R,2y = (5 - m1a)(5 - m2a)(S - m3a) wherein mi, m2, m3 are invariants. Sylvester showed that the quintic might, in general, be expressed as the 5sum of three fifth powers, viz., in the canonical form f— Aq(pf) + k2(qx)5 + k3(rx)5- How, evidently, the ( third transvectant of /, expressed in this form, with the cubic px lxrx is zero, and hence from a property of the covariant j we must have j =pxqxrx ; showing that the linear forms involved are the linear factors of/. We may therefore write f=z Aq(5 - wqa)5 + Aq(S - m2a)3 + &3(S - m3af; and we have merely to determine the constants Aq, Aq, k3. To determine them notice that R, = (a5) and then (/>a5)5 = - R5(*i+K+*3)’ (/.a45)5 = - 5R5(m1A:1 + mjc2 + mjc^), (/, a3o2)5 = - lOR^m^ +m2A;2 + ?n2A;3), three equations for determining A,1, k2, k3. This canonical form depends upon / having three unequal linear factors. When 0 vanishes j has the form j^pfyx, and (/,/)3 = (ap)2(ag')a|=;0. Hence, from the identity ax(pq)—px(aq)-qx(ap), we obtain (,jP(.ap)(a the required canonical form. How, when C = 0, clearly (see ante) R2/ = b2p where 6R4./= B55 + 5B5> - 4A2pB, which is Bring’s form of quintic at which we can always arrive, by linear transformation, whenever the invariant C vanishes. Remark.—The invariant C is a numerical multiple of the resultant of the covariants i and /, and if C = 0, p is the common factor of i and j. The discriminant is the resultant of CCQGy 77 and CX2 and of degree 8 in the coefficients ; since it is a rational and integral function of the fundamental invariants it is expressible as a linear function of A2 and B ; it is independent of C, and is therefore unaltered when C vanishes ; we may therefore take / in the canonical form 6R4/= B<55 + 5B54p - 4A2p5. The two equations ^=5(B54 + 4B53p) = 0, §L=5(B54-4A2p4) = 0, dp yield by elimination of 5 and p the discriminant D = 64B - A2. The general equation of degree 5 cannot be solved algebraically, but the roots can be expressed by means of elliptic modular functions. For an algebraic solution the invariants must fulfil certain2conditions. When R = 0, and neither of the expressions AC - B, 2AB - 3C vanishes, the covariant ax is a linear factor of f; but, when R = AC - B2 = 2AB - 3C = 0, ax also vanishes, and then / is the product of the form /| and of the Hessian of j3. When ax and the invariants B and C all vanish, either A or j must vanish ; in the former case j is a perfect cube, its Hessian vanishing, and further / contains j as a factor; in the latter case, if px,<rx be the linear factors of i, /can be expressed as (pcr)/=c1p^ + c04 i if both A and j vanish i also vanishes identically, and so also does/. If, however, the condition be the vanishing of i, f contains a linear factor to the fourth power.