ALGEBRAIC seminvariant forms established. Putting n equal to co, in a generating function obtained above, we find that the function, which enumerates the asyzygetic seminvariants of degree 0, is 1 -z2.l -2s.1 - 24....! that is to say, of the weight w, we have one form corresponding to each non-unitary partition of w into the parts 2, 3, 4, ...6. The extraordinary advantage of the transformation of 0 to association with non-unitary symmetric functions is now apparent; for we may take, as representative forms, the symmetric functions which are symbolically denoted by the partitions referred to. Ex. gr., of degree 3 weight 8, we have the two forms (322), a(24). If we wish merely to enumerate those whose partitions contain the figure 0, and do not therefore contain any power of a as a factor, we have the generator ~0 If 0 = 2, every form is obviously a ground form or perpetuant, and the series of forms is denotedc+1 by (2), (22), (2s), ...(2K+1) ... Similarly, if 0 = 3, every form (3' 2x) is a perpetuant. For these two cases the perpetuants are enumerated by z2 z* rr^> and i -s2.i -s3 respectively. When 0 = 4 it is clear that no form, whose partition contains a part 3, can be reduced ; but every form, whose partition is composed of the parts 4 and 2, is by elementary algebra reducible by means of perpetuants of degree 2. These latter forms are enumerated bvJ 1 - s2sr.1 - z4a ; hence the generator of quartic perpetuants must be
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of lower degrees if + + ...+<r0ci0 can be broken up into any two portions (oqcq + <72a2 + ... + crsas) + 05+ias+1 + <r4+2as+2 + • • • + 0'6»ae). such that crl + 0'2-K.. + 0'* — 0> f°r then crs+i + avt-2+ ... +00 = 0 ; and each portion raised to any power denotes a seminvariant. Stroh assumes that every reducible seminvariant can in this way be reduced. The existence of such a relation, aso-i + o-a+.-. + a-^O, necessitates the vanishing of a certain function of the coefficients Ao, A3, ...A#, and as a consequence one product of these coefficients can be eliminated from the expanded form and no seminvariant, which appears as a coefficient to such a product (which pay be the whole or only a part of the complete product with which the seminvariant is associated), will be capable of reduction. Ex. gr. for 0 = 2, (oqcq + cr.2a2)'u’; either oq or 0-2 will vanish if 0-^2=A2=0 ; but every term, in the development, is of the torm (222...)A|iW and therefore vanishes; so that none are left to undergo reduction. Therefore every form of degree 2, except of course that one whose weight is zero, is a perpetuant. The z2 generating function is ^ For 0 = 3, (<T1a1 + <rtfi2+<T3a3)w ; the condition is clearly o-i0-2<r3 = A,=0, and since every seminvariant, of proper degree 3, is associated, as coefficient, with a product containing A3, all such are perpetuants.3 The general form is (3*2^) and the generating _ . z function 1_g2>1_^For 0=4, (o'ltti + 0'2a2'h 'h 5 the condition is oqo^o-gcr^aq + 02)(<7i + <J3)(<ri + = A4A3 = 0. Hence every product of Ai, A2, A3, A4, which contains the product A4Ao disappears before reduction ; this means that every seminvariant, whose partition contains the parts 4, 3, is a perpetuant. The general form of perpetuant is (4K3A2'J-) and the generating function
2 4 2 3 4 1 — Z2.1 — Z3.1 — : 'l-z .l-z 1-Z .1-Z .1-Z ’ and the general form of perpetuant is (4'c+13;+12^). When 0^=5, the reducible forms are connected by syzygies 1 — z2.1 — z3.1 — z4* which there is some difficulty in enumerating. Sylvester, Cayley, and MacMahon succeeded, by a laborious process, m In general when 0 is even and =2cf>, the condition is establishing the generators for 0 = 5, and 0 = 6, viz. : ®1®2. • .^n(<ri + <r2)n(<ri + <r2 + <r3).. .HK + <r2 + ••• — and we can determine the lowest weight of a perpetuant; the l-z2.l-z3.l-z4.l-z3’ l-z2.l-z3.l-z4.l-z5.l-z6’ degree in the quantities a is but the true method of procedure is that of Stroh which we are about to explain. ^ , . _ ,. „ Method of Stroh.—hx the section on ‘ Symmetric Functions, Again, if 0 is uneven =20 + 1, the condition is it was noted that Stroh considers <r W ®i®2- • •®2</,+in(°'i + <hs)II(<ri + + o’s) • • •n(°'i + <r2 + ' ■ + 0) — 0 ! (oqaj + oqcta + ... + <r q<xo) > and the degree, in the quantities <r, is where «r1 + <r2+...+ <r0 = O and “? = “?= ... = “*=«, symbolically, to 2*+1+e*+>)+(V(+-+(T) be the fundamental form of seminvariant of degree 0 and weight = 22^ - l = 2e_1 — 1. io ; he observes that every form of this degree and weight is Hence the lowest weight of a perpetuant is 20-1-l, when 0 is linear function of such symbolic expressions. We may write > 2. The generating function 0is 1thus 3 (1 + cr^Xl + cr2|)...(l + <r0?) = 1 + A2f + A3f + ... + A0^. 2 “ -1 2 If we expand the symbolic expression by the multinomial (1-z )(1-33)(1-24)---(1-20) theorem, and remember that any symbolic product a2 “3 The actual form of a perpetuant of degree 0 has been shown by retains the same value, however the suffixes be permuted, we MacMahon to be a71"1 ay2 a™3 v irj ir2 , 4 k +2 1 o0-4 c 3 •••> shall obtain a sum of terms, such asw>! -L<? +1 ,0-2 c0-2 +2 ?-3 te-3+4 3 3 0-l ir1I 7T2! 7T3' ,2^ 1-1 which in real form is w !
- and, if we ex- k , k
...k2 being given any zero or positive integer values. 0 press in terms of A2, A3, ... and arrange the whole ^Simultaneous Seminvariants of two Binary Forms.—Taking the as a linear function of products of A2, A3, ..., eacli coefficient will two forms to be be a seminvariant, and the aggregate of the coefficients will give atfc+paix ^2+0(0 _ l)a2a3l x2 + .•■■-apX2, us the complete asyzygetic system of the given degree and weight. b(fe + qbix ^x2 + q{q-)b2x x2 + ...+b(jc2, When the proper degree 0 is < w a factor a™ must be of course understood. every leading coefficient of a simultaneous covariant vanishes by Ex. gr. the operation of (r a (r a <T a c a 2 S<r + 2 0 02 d ,d.d d_, h 2t( l l + 2 2 + 3 3 + 4 4) — 2^ 1 “l® ^ ’! ’ + bo 4 Oa + 04 = ao^ + «i^-2+.' ■ + dh hdb2 dh. = a2( - 2A2) + a.f A2= {a - 2a2)A2= (2)A2=a.^(2)A2. that we may employ the principle of suffix diminution In general the coefficient, of any product A^A^A^..., will have, Observe obtain ic from any seminvariant one appertaining to a p- 1 ana as coefficient, a seminvariant which, when expressed by paiti- to a q-l , and that suffix augmentation produces a, portion ol a tions, will have as leading partition (preceding in dictionary higher seminvariant, the degree in each case remaining unaltered. order all others) the partition (ttjTt^...). Now the symbolic Remark, too, that we are in association with non-unitary symexpression of the seminvariant can be expanded by the binomial metric functions of two systems of quantities which will De theorem so as to be exhibited as a sum of products of seminvariants, S. I. — 39
