312
ALGEBRAIC
While, in general, 02 V k n-2k /j? jp Yb (n-2k) a x £cf dxl) x 2 2 /0 0 fc 7l! fc TO_2fc V0F?+0^|J (a3a) =rn-2^' " (ti - 2A;)!aa(^) To these results it is useful to add / 02 02 Yan-l, a , (n-l) kan-2k-l ~,a„a.x xa). 5^2 ) axx (W dxi + dx) w )-(n-2lc-iy> Ex. gr. from the quadratic we derive aa = al+al=a0 + a.2, from the cubic the linear covariant aaax = (a0 + a2)a:i + (% + a3)a;2, and from the cubic covariant a^xa) of the cubic aa{xa) = («! + - (a0 + a2)x2, a linear covariant otherwise obtainable by operating upon aaax with 0 0 Xl dx2 *20a;1’ Similarly, from the quartic a^, we obtain the quadratic covariant aaa2x=(a0 + a2)xl + 2(a1 + a3)x1x2 + (a2 + a4)x% , and the invariant a^ = a0 + 2a2 + a4. It has been shown, by Sylvester, that all orthogonal covariants satisfy the partial differential equation a
°fa;+2aifa2+-+nan-lfa„ 3
, 3 ^ 3 _o
b [7b — 1)^2^ rb ... “br Cby
{T_ ' -~ “' fcn_: } + Xldx2 X2dx4 ’ which, in the notation employed in a previous page, is
FORMS
of covariant we have a special differential We can 2 equation. infer from the annihilating operator V(V + 4) for covariants of the second order that the function (Y2 + 4)C0is an invariant of the same binary form. Ex. gr. For the binary quadratic a|, a0 is the leading coefficient of the form itself, a covariant of the second order, and as a consequence the operation /0 0 + 3 a 3 o2ai 3 V + 4„ V “*0^ ~ °0a1" 0^2j when performed upon a0 must produce an invariant; it, in fact, produces the invariant 2(a0 + «2). The annihilator, in respect of a covariant of order 2s, is V(Y2 + 22)(V2 + 42)...(V2 + 4s2); and if the order be 2s +1 (V2+12)(Y2 + 32)...(V2 + 4s2 + 4s + 1); so that, in general, the operation Y2 + e2 upon a leading coefficient of a covariant of order e produces a leader of a covariant of order e - 2 appertaining to the same binary form, and this must necessarily be of the same degree in the coefficients. Ex. gr. From the Hessian of the cubic a2, viz., we thus obtain the invariant V2 “ ~ = ah{aiTIt will be noticed that a source of a linear covariant is annihilated by V2 + l. The Boolian System.—The form is = ^aix > associated with the quadratic cc2 +2cos wa^a^ + a?2. ax is transformed to A^ = (A1a:I + A2a?2)?l, by the substitutions x 1 _sin(to-q)x1 [ sin (to sin w sin w 2 ’ sin a„, . sin x0 = Xi + sm w X2,
Oa-Oa+(a5^)=0. of modulus-^sin w : where w' = B-a. If then a covariant, of order e to an nic, be The problem is to find functions of the original coefficients, 20^ £2, variables, and w, which by the transformation become the like the operation produces functions of the new coefficients and variables and a/, save as to power of the modulus. ax and (ab) clearly possess the invariant X{x[- kx{aa ~ %)Ck + CM"fc+14"1 - C;;(e - k)x{'^X^1}, aproperty. k Put a4 - a2 cos w = b1-b2 cos «=...= cq = /3j =..., the vanishing of which necessitates the relations which, writing a2 — oqcos w = b2-b1 cos w=... = a2=^S2=... Oa-^a=Y, take the form x4 + x2 cos w = , Cj YC0 =0, x2 + x4 cos w = |2, 2C2- VCi eC0 =0, where a
ft, /3.2,... are new umbrae and £1} ^2 are auxiliary 2 3C3-YC2 =0, variables. Let the transformation convert a1, a2 into Aj, A2, where e Ce — VCe_! — 2C6_2 = 0 ; A: = Ax - A2 cos w', and thence A2 = A2 - Ax cos a)', C^VCn,2 and ft into Si, S?2, where 2C2=|(V +2 l) + (6-l)}C0, 51 = X1 + X2cos u' = cos a aij + cos (w — a)x2, 3!C3={V(V + 4) + 3(e-2)V}C0, 52 = X2 + X1 cos a/. = cos^ aij + cos (w - p)x2. 4 !C4= |(V2 +2 1)(V2 +2 9) + 6(e - 3)(Y2+1) +2 3(e - l)(e - 3)}C0, 5 !C5= {V( V + 4)( V +16) + 10(e - 4)V(V + 4) Observe that + 15(e - 2)(e - 4)} C0 ; xg=xxft + Xc£2+ 2 cos wx4x2 + . Now indicating the manner of obtaining the successive covariant cox1=sl >4»1I-ss^k efficients when once the leading coefficient C0 is known. sinw smw 2 sina sin (w-a) Ex. gr. Taking as leading coefficient a|, the order of the coX9=- -—,x-,-
- /.i Xo,
t.y x oin variant is 2n - 4, and the three first terms sin a a , A. x — sinsmw 1(to-a)^1 -|—si"'* («0 + 2«! + cl^x^1 ~ 4 sm to 2 l 5 _ sin (to-ft si sin B + 2 (Vi, — 2)(gi(fii -|- 2ct-4a% + a^a^)x4 x% ——o
- ^2 — ” oirTTr
’’ ■ sm smw to + (n-2')[(n - 2'){a + 2a2 + ag) + {n- 3)(a0ct2 + 2a a3 + a2a4) jw' f , ,,, • , A i~ sin vX x2n-6x2 sm to {cos(w-fta1-cos^a2}, 4 2 sin to' + .... sm w {- cos (w - a)®! + cos aa2}; It may be gathered, from the above relations, that, for covariants of orders 0, 1, 2, 3, 4,..., the leading coefficients C0 satisfy the whence it can be shown that equations (X A) = XjA2 — X2Ax = Xx(A2 — Aj cos w') — X2(Aj — A2 cos w') vc0=o, sin w' . =— (V*2 + 1)C0=0, sm to [xa); 2 V(V2 + 4)C0 = 0, from which we learn that (xa) possesses the invariant property. (V + 1)(Y + 9)Co = 0, Further, it may be verified that Y(V2 + 4)(V2 + 16)C0 = 0, sm a; smco respectively. It is thus important to notice that for each ^der sin o; sin w
