algebraic
forms
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same degree - order. To see how this is take the linear (ii.) in descending order as regards the exponents of by and (ya) ; the ^ 0/0 (iii.) in ascending order in regard to the exponents of ay, {yb), and operator Xl x x bx. We have then the notion of adjacent members, and we can ^~
- dx
1- bx show that the difference between any two adjacent members is either divisible by (ab)(xy), (ab)xy or (ab)yy, and the theory of the and further the operator of order k polar members may be proceeded with in the usual manner. 0 0 k_f 0 fc Xl The process of transvection can be extended in the same way d^~X2dx1) -dx) ’ as the polar process. Of two forms obtained by symbolic expansion. f— ax 1 4* — Ox Then d n n , dc n-k we define the transvectant of orders k, l to be (n-k)v-xa) ax . ' 0 £C / ,<p), /c . Z / 'kfl-jTb = a (f n , .n-k k . 0Y n c (n-k) (xa) ax j dx) {xa) ={-)*rrz = (ab) abax bx so that and similarly
n , di
fam,(xb)^nTc,l / k, isI m-Ic —l, T
fc l %) *x = (xa) > | =(-) ab{ab)ax (xb) , x n {xa) ,bx ' =(-) a (ab) (xa) bx , izl '(^ ({xa)sn =an X
fa) x’ n m k n lc 1 11 n k1 k ' ~xb) ~ ' -, •[ (xaf ,(xb) } ’ — (ab) ab(xa) and d n „ I a„n =a~7 (-)”/ 2 and if the forms he </>™, , (n) ' k l .n-k-l, Yx KtxM =(W) 4>^4>x Hence the operation of writing - x2, xx in place of xx, x2 in any and similarly form of or (xaf is, disregarding a numerical factor, equal to the {x^) , {(xct))m,(xp)n}k,l = { ^{x<P) performance of (a^y1, an operation of order n. Hence, when 9*/ and the intermediate processes also in the same way. Taking as this operation isv performed upon any symbolic product before/= a™ ,</> = we find that the process of transvection of (Kxaf{xb)(1...aSxbtx... order k + l is equivalent to the performance of the differential operation of order n in x, it effectively produces f df d<p 0/ 0<ftfc/ d/ ^ [ d/ 9^ V. afbqx...(xa)s(xb)t... 1 dx1dx2 0aq dx1) ^0*! dx1 cx2 cx2 J The operations omitting a mere numerical factor. The multiplication of opeiations is, of course, symbolic, so that (4),(4)2-(4)” [fcJ dxJ dxldxi We will now give some examples of transvectants. Ex. gr. iif=a%,(p = b%, (f,<f>ro=*yx (/, (p)1’0 = (ab)axbx — {a0b1 - a-fi^xl + {a0b2 - a^xxx^ — + (®1^2 > (f,<p)0,1 = abaxbx=K&o + a AM + (ao^i + Mo + a^+a^x^ + (®i^i + <^2^2)*! > (/*) ^:= + ct^bo, 1 1= —a if > 0) ’ (ttb)ab = a-o&i — ai&o + 2^i > 2 2 {f ,<ff’ ^al=Mo + Mi + M2 > _ five orthogonal concomitants of two simultaneous _ quadratics obtained by simple transvection. If a, b be alternative symbols (/>/,)1’° an^ if)/')1’1 vanish identically, and we are left with the quadratic covariant (f,f,)°’1=i< +a?)£C? +2K“i + M2Ka:2+(ai +al):c2 > and the two invariants (A/)2’0 = 2(a0«2 - «?), ifjT^^l + ia + at. We may proceed by transvection from the forms (xa)ax and (xa)2. Observe that (xa)ax = axxf - (a0 - a2)x1x2 - axx% , (xa)2^a^pc^ — <Zaxxxx2--a$c2 , thus {(asa)2, }1 ’ ° = abixa)bx = (aJ)Q + a2bx)xl - (Mo - a2b2)XiX2 - (a0b1 + af>2)xl ; •[ (xaf, } ° ’1 = - (ab)(xa)bx = - } (etA - a2b0)xl + ( - a0bl + aj>0 + a1b2 - aji-^x^ + (a1b1 — a0b2)xl j j 1 0 {(xaf, (xbf } ’ = (ab)(xa)(xb) = (a-A - aj}x)x - (afb2 + (®o^i" Mo)*2 1 where observe that this form is obtained from (ab)axbx by writing -x2,xx in place of xx, x2, a process which is always invariant. From one covariant we can in this way always derive anct ici 0
when performed upon ax or (xa)n produce the n +1 covariants, of order n, ax,(xa)ax~l,(xafax ",...(xaf •, which we may conceive to be generated by the expansion of {ax + (xa)}n ; that is to say, by giving, in a”®!’®2 the increments + Xa2, - a1, respectively. We have a relation connecting any two quasi-adjacent covariants of the series, for since ax~i~(xa) =a(Xxx, (xaf ~ 2ax ~p+2 + (xa)pax ~p — aa(xa)p ~ 2ax ~ pxx; which shows that the sum of every two quasi-adjacent covariants contains the factor aaxx. The identity obtains whether n be the order of the original form or no. From it may be derived others of the kind (xaf ~ 2ax ~p+2 + 2(xa)pax ~p + (xa)p+2ax ~P~2 — aa2/{xa)p-2(1%n-p-2 2 These relations indicate that these covariants do not constitute a fundamentally irreducible set of covariants ; for the covariant aa(xa)f>'2a1il~pxx, is the product of aa(xa)p~2ax~p, xx each of . tne which is a covariant; and-..•ii similarly aa2/(xa)p-2axn-p-2 xx2 is p 2 p 2 2 product of al(xa) ~ ax' ~ and a , each of which is a covariant. Between the n+1 covariants we can establish ft - 1 independent relations giving u reductions ; the system is therefore reduced to two forms which we may take to be ax and (xa)ax . So also, in regard to any covariant (pX) we need only consider the further form (x<p)<px 0> A process, somewhat similar to transvection, may like be performed upon a single form ; this is _0^ +, _02.=A dxl dx% 0aA dX in analogy with the notation x+x=xx. We have n-2 l n (7a, n 0£C? "I"0a3 x ~ 1.2 ) oP‘x a covariant of degree-order 1, ft - 2.
