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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 ^~ 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 / , =(-) 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 = 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 dro=*yx (/, (p)1’0 = (ab)axbx — {a0b1 - a-fi^xl + {a0b2 - a^xxx^ — + (®1^2 > (f, (/*) ^:= + ct^bo, 1 1= —a if > 0) ’ (ttb)ab = a-o&i — ai&o + 2^i > 2 2 {f , _ 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^ — 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 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.