306
ALGEBEAIC
denoted by partitions in brackets ( )a, ( )b respectively. Solving the equation (fla + = 0, by the ordinary theory of linear partial differential equations, we obtain p + q + 1 independent solutions, of which p appertain to Uau—d, q to = the remaining one is Jas = ao&i~«i&o, the leading coefficient of the Jacobian of the two forms. This constitutes an algebraically complete system, and, in terms of its members, all seminvariants can be rationally expressed. A similar theorem holds in the case of any number of binary forms, the mixed seminvariants being derived from the Jacobians of the several pairs of forms. If the seminvariant be of degrees 9,0' in the coefficients, the forms of orders p, q respectively, and the weight w, the degree of the covariant in the variables will be pd + qd' - 2w = e, an easy generalization of the theorem connected with a single form. The general term of a seminvariant of degrees 6, 9' and weight w will be p q P q Z/35 = 0,2<rs:=0' and 2s/)s +2s(rs = w. xi ii The number of such terms is the number of partitions of iv into 6 + 6' parts, the part magnitudes, in the two portions, being limited not to exceed p and q respectively. Denote this number by (w ; 6,p ; 6', q). The number of linearly independent seminvariants of the given type will then be denoted by {w )6, p •, 6', q) -0-1 ',6, p ■, 6', q) ■, and will be given by the coefficient of a^'zw in 1-2 l-a.l-az.l - as2. ... 1 - az?. I- b .Y — bz. 1 - 6z2 ... Y-bz? ’ that is, by the coefficient of zw in 1 -zp+1. 1 - zp+‘2. ... Y - zp+6. 1 -s?+1.1 - 2?+2. ... i _ zq+9' 1 - 2.1 - 22. 1 - z3. ... 1—20.1-22.l — 23. ... 1-20' 5 which preserves its expression when 6 and p and 6' and q are separately or simultaneously interchanged. Taking the first generating function, and writing azP, bz?, i for a, b, and z respectively, we obtain the coefficient of a^b^'zP9^^ ~ “w, that is of afiWz*, in 1 —2—2 -azp. 1 -a2p_2.... 1 - az_i,+2.1 - az~p. 1 - bzq. 1 -.... 1 - iz~9+2.1 the unreduced generating function which enumerates the covariants of degrees 6, 6' in the coefficients and order e in the variables. Thus, for two linear forms, p = q=Y, we find 1 - z-2 Y -az.Y - az-1 .Y-bz.Y- bz*1’ the positive part of which is 1 . Y - az. Y - bz. Y - ab ’ establishing the ground forms of degrees-order (1, 0 ; 1), (0, 1 ; 1), (1, 1 ; 0), viz. :—the linear forms themselves and their Jacobian Ja6- Similarly, for a linear and a quadratic, p = Y ,q —2, and the reduced form is found to be 1 - a%2z2 1 — az. 1 — fe2.1 - abz. 1 — 62.1 — a26’ where the denominator factors indicate the forms themselves, their Jacobian, the invariant of the quadratic and their resultant; connected, as shown by the numerator, by a syzysy of desreesorder (2,2 ;2). ’ J & The complete theory of the perpetuants appertaining to two or more forms of infinite order has not yet been established. For two forms the seminvariants of degrees 1,1 are enumerated by an d the only one which is reducible is a^bo of weight zero ; hence the perpetuants of degrees 1,1 are enumerated by where
J—1 = _1_; 1 —z Y—z’ and the series is evidently a0&i — aj&Q, a0&2 — a1b1 + a2—&oa> > a 0^3 ~ ®1^2 + ®2^1 'J- Q , one for each of the weights 1, 2, 3, ... ad injin. For the degrees 1, 2, the asyzygetic forms are enumerated by •j-_ ^ 1 _ g2) and the actual forms for the first three weights are
F O E M S
V’o. (a0&i — aj&Q)^, (a0&2 - aj)x + aj)0)b0, ajb- 2b0b2), (a0&3 — aj)^ + aj>x — a^b^jb^, ffi)(^1^2 ~ ^o^s) — ai(^i — 2&0&2) j amongst these forms are included all the asyzygetic forms of degrees 1, 1, multiplied by 50, and also all the perpetuants of the second binary form multiplied by a0 ; hence we have to subtract from the generating function 1 and 5, and we obtain the 1—z 1-z2 generating function of perpetuants of degrees 1, 2 1 2 1 z2 2 _ z3 2 1-Z.l-Z 1-2 1 - Z ~ 1 — 2.1 - 2 " The first perpetuant is the last seminvariant written, viz. :— ffi)(^o^2 — Sb^bj) — ax(b — 2bj)j), or, in partition notation, a o(21)&- (l)a(2)j; and, in this form, it is at once seen to satisfy the partial differential equation. It is important to notice that the expression {6)a{6'V)b - {6Y)a{6'V^)b + {6Y2U6'Y*~2)b - ... + {dY%{6jb denotes a seminvariant, if 6, 6', be neither of them unity, for, after operation, the terms destroy one another in pairs ; when 0 = 0, (0)a must be taken to denote a0 and so for 6'. In general it is a seminvariant of degrees 6, 6', and weight 6 + 6’ + s-, to this there is an exception, viz., when 0 = 0, or when 0' = 0, the corresponding partial degrees are 1 and 1. When 6 = 6'—0, we have the general perpetuant of degrees 1, 1. There is a still more general form of seminvariant; we may have instead of 0, 0' any collections of non-unitary integers not exceeding 0, 0' in magnitude respectively. Ex. gr. (2A23A3... 0A0)a(l V^3. •. 0'/^)& - (l2A23Aii.. 0A0)a(ls_ V23M3...0,,V)& + (lV23 A3- • • 0Ae)a(ls “ V^3... 9,IJ'e')b ( - )s(lS2A23A8...0A0)a(2,i23,i3...0'M^)6, is a seminvariant; and since these forms are clearly enumerated by 1 1 -z. 1 -Z2. ... 1 -20. 1 -Z2.1 -23. ... 1 -20' ’ an expression which also enumerates the asyzygetic seminvariants, we may regard the form, written, as denoting the general form of asyzygetic seminvariant; a very important conclusion. For the case in hand, from the simplest perpetuant of degrees 1, 2, we derive the perpetuants of weight w, a0(21“’-2)j - a1(21“'-3)i + a2(21“’-4)j - ... ± aw-2(2)b, 2 4 a0(2 l— (6 - a1(22l“_5)5 + a2(22l»-<% - ... ± aw_4(22)6, ®0(23lw-6)d - a1(23lw-7)6 + a2(23l“’-8)j _ ... + aw_6(23)6, a series of jjjM - 2) or of ~ 1) forms according as w is even or uneven. Their number for any weight w is the number of ways of composing w-3 with the parts 1, 2, and thus the generating function is verified. We cannot, by this method, easily discuss the perpetuants of degrees 2, 2, because a syzygy presents itself as early as weight 2. It is better now to proceed by the method of Stroh. We have the symbolic expression of a seminvariant, + <r a
2 2+... + <rea0 + D/3! + T2I82 + ... + Te,pe,)w
where _1=_2_ ..=a,;-l=-|=... = &; s! s! s! s! and l + <r2 + ••• + <r0 + rl + l"2+ ... +Tq = § . Proceeding as we did in the case of the single binary form we find that for a given total degree 0 + 0', the condition which expresses reducibility is of total degree 20+e,-1-l in the coefficients tr and r; combining this with the knowledge of the generating function of asyzygetic forms of degrees 0, 0', we find that the perpetuants of these degrees are enumerated by z220+0'-l — 1, 2 3 l-z.l-z .! —Z . ...],-20. f-Z2.!-23. ...1 -20'’ and this is true for 0 + 0'= 2 as well as for other values of 0 + 0 (compare the case of the single binary form). <r
