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ALGEBRAIC
From Q|Q2/ = (aA)a|A2/, we obtain, substituting + A2, - for yz, (QA)Q2xAx = (aA)(AA)a2xAx, or (Q, A)1 = i(AA )a|{ (trA)A^ - (aA^A^.},
FORMS
When, on the other hand, a0, cq,... are functions of a0, q,... the matter is not so simple, for now (5*) an operator of order k is not equivalent to 6*, k successive operations of 8. Write
- then
S2 = (<52) + Sx+Sx when SxfSx is an operator formed by operating upon Oj with 5x regarding the former merely as a function of a0, ax,... ; so that 5a Si+5i = (51ao)^K da^ da + ( i i)^-+--n
= _i(AA>4=-|R./; ••• Q/+aq=q(i+^2r) -|R/(X+^3R)’ = (i+1x=k)(q-1xk./). So also it may be shown that R/+AQ = R + R2 + JR3.
write 5xt5i = 52, and generally S1t5,=5*4.x, then it is easy to show that 2](5x) —2l(5x “ 5a)> 3l(5P =
“ 35152 + 253)>
according to a well-known law established by the theorem of operator-transformation given in the section on symmetric functions. In general, if= be any two forms, and any invariant The Partial Differential Equations.—It will be shown later of f be j — 0{a0, a^, the corresponding invariant of f+(p, that covariants may be studied by restricting attention to the leading coefficient, viz., that affecting where e is the order of the covariant. An important fact, discovered by Cayley, is jf+xq, — 0(ao + ^a0) ai + ^al> ■••am + ^m) that these coefficients, and also the complete covariants, satisfy = euj, certain partial differential equations which suffice to determine where them, and to ascertain many of their properties. These equations 3 + — 9 + - • • +-- 0 --. can be arrived at in many ways ; the method here given is due o* —-ciq—=OCt-^ CCf'jYi to Gordan. Xj, X2, /q, /q being as usual the coefficients of subNow, if j be of degree v in the coefficients a, it is a homogeneous stitution, let function, and may be denoted symbolically by Xl + XaJ-^D j 0^ + X20^-I)^ • , xJ^ Vl 0/*2 A/a Pa=(P0a0 +P1a1 + - +Pmam^ Ml0Xx + /A2X ~ DMA ’ ^g^q + involving to +1 umbrse. “ DW* ’ 2 Thence be linear operators. Then if j, J be the original and transformed forms of an invariant = eXSj = {'Pa + ^PaT = 2’ 3 = {y.)wj, k w being the weight of the invariant. and, comparison of the coefficients of*, gives Operation upon J results as follows :— k ^)j = Ck)pa ' Pa = CkXpDa = (fc)ia’ •DaaJ=—o; which is simply a numerical multiple of the kth a - polar of j. & = 0 ; D^J = wJ. It must be noticed that (5 ) denotes an operator of order &, and is a homogeneous that it is only equivalent to & successive operations of 5 in the The first and fourth of these indicate that particular case when the coefficients a0, cq,...are independent of function of X2, and of /q, /t2 separately, and the second and arise from the fact that (Xyx) is caused to vanish by both the coefficients a0, a,,.... The operation of d upon a symbolic third product is very simple ; suppose A to be a symbol which occurs Dv and D^. in the product; it may present itself in a determinant factor say Since J = F(A0, Ax,...A*,...), where A* = aA ka!^, T (Ap), or in a power form A^. If w e find oA™=i//™ we write ^ for A we find that the results are equivalent to wherever A occurs, and thus obtain one term in the result of the operation ; the complete result is obtained by summation in regard to all the symbols dealt with in this manner. The operation of S upon a transvectant, expressed as such, is precisely similar. Ex. gr. Let Fj^ (a6)2(ac)2(Jc)2, 2 k k where /=4. 0=<4> According to the well-known law for the changes of independent and Sax — 8bx = 8cx = ax ; variables. Now then D A 2 2 2 2 2 2 2 2 2 AA fc=(2l-*)Afc;DAMAfc = A:Afc-l; * 5FX= (a&) (ac) (5c) + (aa) (ac) (ac) + (a6) (aa) (5a) , D A = 3(a6)2(aa)2(&a)2. ^A fc = (^-^)Afc+i ;DwAfc = *Afc ; Again let F2= {(/,/')2,/'}4, so we obtain where Sf= Sf = 8f" = 4>; 0J then 2 4 5F2= 2 4 {(/, 0) /'} + {( f,f?, <P} = 3{(/,/') ,0} . So far we may always go whatever the values of a0, aj... ; but —fc+l0ljfc 3J -O _A ; ^ _*U 0Jk — wJ; (rx- /t)A 2jkK when these are independent of a0, cq, ...we may Introduce new 2 symbols and obtain k k 5% = 3(/%)22(/3a)22(6a)22+ 3(a/3)2(aa)2(/3a)2, equations which are valid when Xj, X2, /Xj, /x2 have arbitrary values, = 6(a/3) (aa) (ai8) ; and therefore when the values are such that J =j, Ak = ak. 53F1-6(7i3)2(7a)2(ai3)2 ; Hence XS so that e Fi naQ-l- + {n+ {n- 2)a^~- + ...=wj, 0ao oa1 ca2 = {ab)ac)bc)2 + 3(a&)2(aa)22(&a)X2 2.2 2 2 2 3 + 3(a/3)>a) (a/3) X + (/327) (a7) (aS) X . 2 2 a0^L + 2aiffL+3a2£L+ ...=0, Also 5 F2 = 3l(0',/')2 , <!>}* 4 + ${(/, <p') , 4>V, (jCL-^ C/^2 (jCCq 3 = 6|(^>, 0') ,/} ;4 S F.2=6{(0, p'Y, 0"} ; na n 2 i^ ca0 + («- 1 ja-iMca1 +( ~ ^3Wca2 + • • ■ = o , leading to AS 2 4 2 e F2= {(/,/) ,/'} + 3{f/,/) , 4>V -ai~-0i ‘ +. 02a- q-cj - dj _tvj. , 2C/(l'2 + 3%-=- + + 3{(^, <P'f /}4X2+ {(^, ^)2, ^"}4X3. CGt'i
