ALGEBRAIC
309
FORMS
2 2 2 2 2 showing that, in the present theory, aa, ab, and {xa)_ possess the (a&) by a = a + 2a + a since the identity aabh-a — {abf shows invariant property. Since x+xl=xx we have six types oi the syzygetic relation _ 2 symbolic factors which may be used to form invariants and co(a 0 + a2) -(al+2a+a) = 2(aoa2 ~ «?)• variants, viz.— There is no linear covariant, since it is impossible to form a (a&), aa, ab, (xa), ax, xx. symbolic product which will contain x once and at the same time The general form of covariant is therefore appertain to a quadratic, (v.) is the Jacobian ; geometrically it 3 denotes the bisectors of the angles between the lines os2, or, as ve (ab)ac)bC) • .cfcb^.. .afaX -• • may say, the common harmonic conjugates of the lines a2 and the x(xa)xb)xc)..ax^cx3-^ lines xx. The linear invariant aa is such that, when equated to = (AB)H AC)7l2(BC)713. • • AX’cJ- • • KAcBc-• • zero, it determines the lines as harmonically conjugate to the fcl fc2 fc3 x (XA) (XB) (XC) . • • A|B|C|. • -X™. lines xx ; or, in other words, it is the condition that a2 may denote ic lines at right angles. If this be of order e and appertain to an n To resume the general discussion we recall the relations 2& + 2l + 2m = e, =n A1 = Xa1-i-/aa2> A2= -/iai + ^«2> h + /<2 + ■ • • + 2i +j +j'2 + ... + ki + h = n>
hi + h3+... -tSfg +ji +js+ + k‘2+h i and put X = Xi=:M2; =M = ^2= -^1 > h% + h3+ ... +2is +j2 +js + • • • + ^3 + ^3 = ^ > so that Ax^ga- Xa A2 = <V=( )> viz. the symbols a,b,c,... must each occur n times. It may denote giVinS a simultaneous orthogonal invariant of forms of orders ni,n<i,ns,...; Ak = A2=^ the symbols must then present themselves nx, %2, 7i3... times reM = (^) a)k = - fc(Xa)fc spectively. The number of different symbols a, &, c,... denotes the decree 9 of the covariant in the coefficients. The coefficients = A^=(aX1 + { («m)X1 + gs^X2 } of the°covariants are homogeneous, but not in general isobanc w functions, of the coefficients of the original form or forms. Ol = { Gi^Xj + (Xa)X } = {(ay )X1 + (Xa)X2 } ; 2 the above general form of covariant there are important transformations due to the symbolic identities : four forms of expression. . , , v ^ The polar process is available here to an enhanced degree, t or (aft)2 = tt0bij - a2 ; {xa?)1 — aaxx - ; put n f=a™,ct>=(xa) , as a consequence any even power of a determinant factor may be expressed in terms of the other symbolic factors, and any uneven we not only have m-k k power may be expressed as the product of its first power and a J y — ax ay> function of the other symbolic factors. Hence m the above but also general form of covariant we may suppose the exponents <t>l=(xa)n-ya)k; h, ^2> b-z,...Tc, &3,... converts ax if the determinant factors to be, each of them, either zero or and just as the substitution of xx + ti/^ x2 + ty2 for unity. Or, if we please, we may leave the determinant factors into ax + tay, the same substitution converts (a;a) into (xa) + t{ya), untouched and consider the exponents j, jz, jsyh, h, h, -- to and the symbolic power {(xa) + t(ya)}n be, each of them, either zero or unity. Or, lastly, we may leave the exponents h, k, j, l untouched and consider the product may be regarded as the generator of polars of (xa)n. Note also anShb? u c 3 xm the results a b c ••••Lc >
to be reduced either to the form glg where gr is a symbol of the series a, b, c,... or to a power of xx. To assist us in handling the symbolic products we have not only the identity (ab)cx + (bc)ax + (ca)bx=0, but also (ab)xx + (bx)ax + (xa)bx=0, (ab)ac + (bc)aa + (ca)ab = 0, and many others which may be derived from these in the manner which will be familiar to students of the works of Aronhoid, Clebsch, and Gordan. Previous to continuing the general discussion it is useful to have before us the orthogonal invariants and covariants of the binary linear and quadratic forms. For the linear forms cioaq + axx2 = ax = bx there are four fundamental forms, _ _ (i.) ax = a(fc1 + a1x2 of degree-order (1,1), (.) xx—x+xl ,, (0,2), (iii.) (xa) = alxx-ayx2 ,, (1,1) > (iv.) ab = al +a ,, (2,0), (iii.) and (iv.) being the linear co variant and the quadrinvariant respectively. Every other concomitant is a rational integral function of these four forms. The linear covariant, obviously t e Jacobian of ax and xx, is the line perpendicular to a^, and the vanishing of the quadrinvariant ab is the condition that ax passes through 'one of the circular points at infinity. _ In general any pencil of lines, connected with the line ax by descriptive or metrical properties, has for its equation a rational integral function ot the four forms equated to zero. _ _ For the quadratic ayx +2a1xlx2Jr a2x, we have (i.) a| = a1a:2 + 2a1a:1a:2+a2a:2, (ii.) xx = x+x , (iii.) (a&)2 = 2(a0a2-a5), (iv.) aa=a0 + a2, (v.) (xa)ax — axx + (a2 - a^x^ - a^x^ . This is the fundamental system ; we may, if we choose, replace
1 where
d d / d__(± . ^1dx2 ^'idxl ydx) dx^) ’ indicating processes analogous to the polar process by which a™ is converted into a™ l(ya)k, (xa)n is converted into (xa)n ay. Combining the processes we find that we are in possession of a process, equivalent to a compound partial differential operation, byJ which gs!1 can be converted into ^ ya) > an(i 0118 i • , / .n-ki-lty, Jci k2 n also by which (xa) can be converted into (xa) (ya) ay . We already know that the polar os™ ay satisfies the partial differential equation of the2 second 2order 0m d u _q _ dxxdy2 cx2cyl and the performance of the operation 1 / 02 o2 (m - k)k V dxxdy2 dx2dy0 has been termed the 0 process. As regards the new forms ux—ax (yu) , ^2—(xa) ay, since = (7/1 - k)ka m-k-l (y^) k-l. dxrfyi m lc lc 1 ®2mi —(m-k)ka [fit ft J/VUiy- ~ ~^(va) ~ (— aya-i), dx2dy 02w2 / 7Te U-fc-l a a a„ y x 2 Cx1?nyl = (n- k)k(xa) 2 0 a2 = (n- k)k(xa)n k lak X( - “1*2), 0*202/2
