ALGEBRAIC We may equally express the result as
, or as n(a* - ft)=o. This expression of R shows that, as will afterwards appear, the resultant is a simultaneous invariant of the two forms. The resultant being a product of mn root differences, is of degree mn in the roots, and hence is of weight mn in the coefficients of the forms ; i.c., the sum of the suffixes in each term of the resultant is equal to mn. Resultant Expressible as a Determinant.—From the theory of linear equations it can be gathered that the condition that linear equations in p variables (homogeneous and independent) may be simultaneously satisfied is expressible as a determinant, viz., if allxl + al2x^ + ... + alpxp=0, ^21*^1 "b ^22*^2 "b • • • "b a^pXp 0, appe^ -f-f-... -f- aptfCp—0, be the system the condition is, in determinant form, • •app) — 0 ; in fact the determinant is the resultant of the equations. Now, suppose /and cp to have a common factor x — 7, /j and being of degrees m -1 and n-1 respectively ; we have the identity <p-l(x)f(x) =f1(x)<p(x) of degree m + n-1. Assuming then to have the coefficients Bj, B2, ...B„ and/j the coefficients Alf A2i ...Am, we may equate coefficients of like powers of x in the identity, and obtain m + n homogeneous linear equations satisfied by the m + n quantities Bj^, ...Bn,A1,A2,...A„1. Forming the resultant of these equations we evidently obtain the resultant of/ and <j>. Thus to obtain the resultant of /= a0x3 + a-pn? + ape + «3, <p—bpt? + x + 62 we assume the identity (B0£c + B1)(a0*3 + apr? + ape + a3) = (A0»2 -f Kpc -f A2)(&0a:2+bpe + 52), and derive the linear equations Bq^o ~-A-o^o Bfl*! "b Bjttg — Aq&j — Aj&q — 0, Bo®2 + B1«i — Aq&2 — AjJj — A260—0, "b Bjtt2 — A |/v2 — A/1 = 0, A2b2=0, and by elimination we obtain the resultant ct9 0 0 0 (■?] O'Q b 2 Lq 0 a numerical factor a2 a1 b2 6] bs being disregarded. a3 a2 0 b2 b: 0 a3 0 0 bf This is Euler’s method. Sylvester’s leads to the same expression, but in a simpler manner. He forms n equations from / by separate multiplication by a:”-1, £cn~2, ... mx, l 1, in succession, and similarly treats with m multipliers x ~ , xm~2, ...x, 1. mFrom these m+n equations he eliminates the m + n powers a; +n-1, xm+n~2, ...x, 1, treating them as independent unknowns. Taking the same example as before the process leads to the system of equations «o»4 + ap3 + apx? -f ap =0, ap? + ap»? + aps+a3 =0, x bp + bpt? + b2 x? — 0, b3x?+ bpx?+ b2x =0, 2 bQx +b1x+b2 =0, whence by elimination the resultant «0 “l “2 «3 0 > 0 an a, a»
Zq &2 0 0
0 &0 b2 0 0 0 &0 Zq b2 which reads by columns as the former determinant reads by rows, and is therefore identical with the former. Bezont’s method gives the resultant in the form of a determinant of order m or n, according as m is fn. As modified by Cayley it takes a very simple form. He forms the equation f(x)<t>(x') -f(x')<p(x) = 0, which can be satisfied when /and </> possess a common factor. He first divides by the factor x — x', reducing it to the degree m- 1 in both x and x' where m>n he then forms m equations by equating to zero the coefficients of the various powers of x';
FORMS
281 2
1 1
these equations involve the m powers x°, x, x , ... x” * of x, and regarding these as the unknowns of a system of linear equations the resultant is reached in the form of a determinant of order m. Ex. gr. Put (ap? + ape2 + ape + a^ibp'2 + bpe' + b2) - (a0a;'3 + ape'2 2 + ape' + a3) (bp + bpc + b2) — Q ; after division by x-x' the three equations are formed a^bp22 -f a0bpc + a0b2 — 0, a^bpe2 + {.aQb2 + a1&1 - a2&0)x + alb2- azbQ = 0, aQb2x + («x62 - a3b0)x + a2b2 - a3b1 = 0, and thence the resultant a a > o (J 2 ®0^2 "b ~ ®2^0 ^1^2 — ®o^2 ^ 1 — a^Q a2b2 — which is a symmetrical determinant. Case of Three Variables.—In the next place we consider the resultants of three homogeneous polynomials in three variables. We can prove that if the three equations be satisfied by a system of values of the variable, the same system will also satisfy the Jacobian or functional determinant. For if u, v, w be the polynomials of orders rn, n, p respectively, the Jacobian is (wj v2 and by Euler's theorem of homogeneous functions xu^ + yu2 + zu3 = mu xvj +yv2 +zv3 =nu xw^ + yw2+zw3~pw •, denoting now the reciprocal determinant by (Uj V2 W3) we obtain 3x=muJl + nv'V1+pw'W', 3y=..., Js=..., and it appears that the vanishing of u, v, and w implies the vanishing of J. Further, if m = n=p, we obtain by differentiation / SU, 3Y, + W0Wj J + x0J -7ox 1 + u1U1 + rjVj + ^iWi Y ox ex— — m ( - cx
- S:=(m-1)J + mV‘te+Vte+,°te I’
0J to vanish in this case; Hence the system of values also causes ^ and by symmetry and ^ also vanish. The proof being of general application we may state that a system of values which causes the vanishing of k polynomials in k variables causes also the vanishing of the Jacobian, and in particular, when the forms are of the same degree, the vanishing also of the differential coefficients of the Jacobian in regard to each of the variables. There is no difficulty in expressing the resultant by the method of symmetric functions. Taking two of the equations axmn + (by -fcz)a;m-1 + ... = 0, a'x + (b'y + c'z)xn~1 = 0, we find that, eliminating x, the resultant is a homogeneous function of y and z of degree mn ; equating this to zero and solving for the ratio of y to 2 we obtain mn solutions ; if values of y and z, given by any solution, be substituted in each of the two equations, they will possess a common factor which gives a value of x which, combined with the chosen values of y and z, yields a system of values which satisfies both equations. Hence in all there are mn such systems. If, therefore, we have a third equation, and we substitute each system of values in it successively and form the product of the mn expressions thus formed, we obtain a function which vanishes if any one system of values, common to the first two equations, also satisfies the third. Hence this product is the required resultant of the three equations. Now by the theory of symmetric functions, any symmetric functions of the mn values which satisfy the two equations, can be expressed in terms of the coefficient of those equations. Hence, finally, the resultant is expressed in terms of the coefficients of the three equations, and since it is at once seen to be of degree mn in the coefficient of the third equation, by symmetry it must be of degrees np and pm in the coefficients of the first and second equations respectively. Its weight will be mnp (see Salmon’s Higher Algebra, 4th ed. § 77). The general theory of the resultant of k homogeneous equations in k variables presents no further difficulties when viewed in this manner. The expression in form of a determinant presents in general considerable difficulties. If three equations, each of the second degree, in three variables be given, we have merely to eliminate the six products a;2, y2, z2, yz, zx, xy from the six equations 0J = g-0J = 0J _ if . „ we apply . the , same process to , these ,, tt = v=xij = gg- =0; equations, each of degree three, we obtain similarly a deterS. I. — 36
