A4 (Z) =I 1 —aez12
—az*.—a2.—aez*.1—a8. l —a2ze
establishing the 5 ground forms and the syzygy which connects them.
The process is not applicable with complete success to quintic and higher ordered binary forms. This arises from the circumstance that the simple syzygies between the ground forms are not all independent, but are connected by second syzygies, and these again by third syzygies, and so on; this introduces new difficulties which have not been completely overcome. As regards invariants a little further progress has been made by Cayley, who established the two generating functions for the quintic
1—aS6
1—a*.1—a8.1—a12.1—a1e'
and for the sextic
1—a3° 1—a2.1—a4.1—a6. l —alo. l —alb'
Accounts of further attempts in this direction will be found in Cayley's Memoirs on Quantics (Collected, Papers), in the papers of Sylvester and Franklin (Amer. J. i.iv.), and in Elliott's Algebra of Quantics, chap. viii.
Perpetuants.—Many difficulties, connected with binary forms of finite order, disappear altogether when we come to consider the
Stroh assumes that every reducible seminvariant can'in this way be reduced. The existence of such a relation, as al+a2+...+as=0, necessitates the vanishing of a certain function of the coefficients A2, As,...Ae, and as a consequence one product of these coefficients can be eliminated from the expanding form and no seminvariant, which appears as a coefficient to such a product (which may 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, (vial+eaa2)1'; either al or as will vanish if ala2=A2=o; but every term, in the development, is of the form (222...)A2" 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 generating function z2
is
I—z2'
For 0= 3, (vial+a2a2+a3a3)1O ; the condition is clearly ala2a3 = As = 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 (3K2) and the generating function az a 1—z.1—z
For 0=4, (alai+a2a2+asaa+a4a4)1D; the condition is  0'le20'30'4(al+a2) (01+a3) (a1 +0'4) = A4A3 = 0.
Hence every product of Al, A2, As, A4, which contains the product A4A3 disappears before reduction ; this means that every seminvariant, whose partition contains the parts 4, 3, is a perpetuant. The general form of perpetuant is (4K3A20.) and the generating function
z7
638
form of infinite order. In this case the ground forms, called also perpetuants, have been enumerated and actual representative seminvariant forms established. Putting n equal to oo , in a generating function obtained above, we find that the function, which enumerates the asyzvgetic seminvariants of degree 0, is
1
1—z2.1—z3.1—z4....1—ze
that is to say, of the weight w, we have one form corresponding to each nonunitary partition of w into the parts 2, 3, 4,...8. T
e
extraordinary advantage of the transformation of S1 to association with nonunitary 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
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 enumer
a
ated by I —z2. I —z4; hence the generator of quartic perpetuants
must be
z4
1 —z2.1—z3.1—z4 1—z2.1 —z4 =1—z2.1—z3.1—z4' and the general form of perpetuants is (4K+1 3A+1 2').
When 0 5, the reducible forms are connected by syzygies which there is some difficulty in enumerating. Sylvester, Cayley and MacMahon succeeded, by a laborious process, in establishing the generators for 8=5, and 0=6, viz.:
.Lib
1—z2.1—z3.1—z4.1—z5' 1 —z2.1—z3.1—z4.1 —z'. 1 —z"'
but the true method of procedure is that of Stroh which we are about to explain.
Method of Stroh.—In the section on " Symmetric Functions," it was noted that Stroh considers
(a1a1+a2a2+... +aeae)10,
a e s
al a2 a
where al+a2+...+ae=0 and s!—s!==sl=as symbolically, to be
the fundamental form of seminvariant of degree 8 and weight w; he observes that every form of this degree and weight is a linear function of such symbolic expressions. We may write
(1+6,10 (1+a2I)...(1+0.et)=1+A2i;2+A3t3+...+Age.
If we expand the symbolic expression by the multinomial theorem, and remember that any symbolic product ai 1a22a23... retains the same value, however the suffixes be permuted, we shall obtain a
+rl n2 >2. The generating function is thus 
z2~1—1
(1—z2) (1—z3) (1—z4)... (1—ze)
The actual form of a perpetuant of degree 8 has been shown by MacMahon to l?e
(0Ke+1'B ^ Ke_1+1 8 2Ke_2i2 3K9_3+4 ...3KS+204r2K2) Ko,KB_1,...K2 being given any zero or positive integer values. 
Simultaneous Seminvariants of two Binary Forms.—Taking the two forms to be
voxi+palxl' x2+p(p—1)a2xrx2+... +a5xz, box'+gbixi?x2+q(q—1)b2xi2x2 { ...+bQx2,
every leading coefficient of a simultaneous covariant vanishes by the operation of
ila+itb=ao l+alga +...+aP1dap+bodl71+b1db+...+bsldb
Observe that we may employ the principle of suffix diminution
to obtain from any seminvariant one appertaining to a (p—i)'° and
a q—1 *°, and that suffix augmentation produces a portion of a
higher seminvariant, the degree in each case remaining unaltered.
Remark, too, that we are in association with nonunitary sym
metric functions of two systems of quantities which will be denoted
by partitions in brackets ( )a, ( )b respectively. Solving the equation
(Sla+sib) u=0,
by the ordinary theory of linear partial differential equations, we obtain p+q+1 independent solutions, of which p appertain to it.au = 0, q to SZbu = 0;the remaining one is Jab =aobi —a1bo, 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 degree 0, 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 pB+qe' — 2w = s, an easy generalization of the theorem connected with a single form.
ze
1—z2.1—z3.1—z4....1 —zee
If 8=2, every form is obviously a ground form or perpetuant, and the series of forms is denoted by (2), (22), (23),... (2K+1) .. Similarly, if 0=3, every form (3K+12A) is a perpetuant. For these two cases the perpetuants arc enumerated by
z2 z3
z4 z7
zal
sum of terms, such
The general term of a seminvariant of degree 0, 0' and weight w will be
aP0ap1ap2...aPPb°ob°'b°2... b°q
0 1 2 Q 0 1 2
? q ? q
where p,=O, Ev,=O' and sp,+Esc. =w.
1 1 1 1
The number of such terms is the number of partitions of w into 0+0' parts, the part magnitudes, in the two portions, being limited not to exceed p and q respectively. Denote this number by (w; 0, p; 0'. q). The number of linearly independent seminvariants of the given type will then be denoted by
(w; 0, p; 0', q)—(w—1; 0, p; 0', q);
and will be given by the coefficient of aebe'zw in 1 —z
1— a. 1— az.1 —az2. ... 1 — azP. 1— b. 1— bz. 1— bz2... 1— bz2'
that is, by the coefficient of z'° in
1 —zP+1. 1 —zP zP4'1—zP+9. 1 —zq+1. 1—zq+2.... 1—z2+91
1—z. 1—z2. 1—z3.... 1—z9. 1—z2. 1—z3.... 1—z9, ; which preserves its expression when 0 and p and 0' and q are separately or simultaneously interchanged.
Taking the first generating function, and writing azP, bzq, tie for a, b and z respectively, we obtain the coefficient of
aebe'zP9+28'2w. that is of aebe'z', in 1z2
1 azn.lazP z....l az P+2. 1az P.1bzq. 1bzq2....1bz0+2.11,zq
the unreduced generating function which enumerates the covariants of degrees 0, 0' in the coefficients and order o in the variables. Thus, for two linear forms, p=q=1, we find
1—z2
1—az. 1—az 1. 1—1,z. 1—bzl'
the positive part of which is
1
1—az. 1—bz. 1—ebb'
establishing the ground forms of degreesorder (1, o; i), (o, 1; I), (1, 1; o), viz :—the linear forms themselves and their Jacobian Jab. Similarly, for a linear and a quadratic, p=1, q=2, and the reduced form is found to be
1 a2b2z2
1—az. 1—bz2. 1—abz. 1—b2. 1—alb'
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 syzygy of degrees
order (2, 2; 2).
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, I are enumerated by
1 , and the only one which is reducible is ao 0 of weight zero;
hence the perpetuants of degrees 1, I are enumerated by
1
1 z—1=1zz'
and the series is evidently
abbe —a1b1,
ao 2 — a1b1 +a2b0, aoba — a1b2+a2b1— aabo,
one for each of the weights 1, 2, 3,..ad infbn.
For the degrees 1, 2, the asyzygetic forms are enumerated by
1—z.11—z2' and the actual forms for the first three weights are
aubzo,
(a0b1—albo) bo, (acb2 —a1b1+a2bo) bo, a0(bi 2bob2)
(a0b3—a1b2+a2b1—a3bo)b0,
a0(b1b2 3bob3) —a1(bi — zb0b2) ;
amongst these forms are included all the asyzygetic forms of degrees
1, 1, multiplied by b0, and also all the perpetuants of the second
binary form multiplied by ao; hence we have to subtract from the
generating function —z and z2
g 1 1 —z2' and obtain the generating
function of perpetuants of degrees 1, 2.
1 1 z2 _ z3
1—z. 1—z2 1—z 1—z2 I— z. 1—z2'
The first perpetuant is the last seminvariant written, viz.:—a0(b0b2—3b0b3) —a1(b} 2bob2),
or, in partition notation,
ao(21)b — (1),,(2)1,;
and, in this form, it is at once seen to satisfy the partial differential equation. It is important to notice that the expression
(9)„(0'1')1, — (01)a(0'1a1)1,+ (812).(0'1,2)b — ... t (01'),,(0')1,
denotes a seminvariant, if 0, 0', be neither of them unity, for, after,
639
operation, the terms destfoy one another in pairs: when 0=0, (0)a must be taken to denote ao and so for 0'. In general it is a seminvariant of degrees 0, 0', and weight 0+0'+s; to this there is an exception, viz., when 0=o, or when 0'=0, the corresponding partial degrees are 1 and 1. When 0=0'=o, we have the general perpetuant of degrees 1, I. There is a still more general form of seminvariant; we may have instead of 0, 0' any collections of nonunitary integers not exceeding 0, 0' in magnitude respectively, Ex. gr.
(222313...0") a(1'2423p3...0'pe') b —(12a23A3..Oxo)a(1s i2p23p3...0'pe')b +0.221,23 73 ...OAB) a ('22µ23p3..8'µ9') b
(—)'(1'2A23A3...eo)a(2 3µ3...0''10')b,
is a seminvariant; and since these forms are clearly enumerated by
1
1—z. 1—z2.... 1—ze. 1 —z2. 1—z3....1—ze''
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,
ao(21ir2)b `a1(21vr3) b +a2(21w4)1, ... *aw2(2)br •a0(121w4(ba1(221w6)1,+a2(221w6)1, ... taw4(22)1,,
a0(231w2)bal(221w2)1,+a2(231" 0)1,... t aw.6(23)br
a series of 2(w—2) or of 2(w—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.
1(aial+e2a2+ ... +r0a9+r1F91+r2II2//1~+ ... +r0rl+q o,)1O where
a a 4 i3a
s! s!
01+e2+...+v9+T1+r2+... +T9=O.
Proceeding as we did in the case of the single binary form we find that for a given total degree 8+0', the condition which expresses reducibility is of total degree 2e"'11 in the coefficients v and r; combining this with the knowledge of the generating function of asyzygetic forms of degrees 0, 8', we find that the perpetuants of these degrees are enumerated by
z20+0'1_1
1 —z. 1 —z2. 1—z3.... 1—z°. 1—z2. 1—z3.... 1—ze"
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).
Observe that, if there be more than two binary forms, the weight of the simplest perpetuant of degrees 0, 0', 8",... is 2e+e'+e"+ — 1, as can be seen by reasoning of a similar kind.
To obtain information concerning the actual forms of the perpetuants, write
(1 +vix) (1+2x)... (1 +vex) = 1 +Aix+A2x2+... +Aexe (1 +rix) (1 +72x)... (1 +r5'x) = 1 +Bix+B2x2+... +B0'x°' where Ai+Bi =O.
For the case 0=1, O'=1, the condition is
v1r1 =A1BI = 0,
which since Ai+Bi=o, is really a condition of weight unity. For w=1 the form is Aiai+Blb1, which we may write aobi—aib0 =
a0(I)b(I)abo; the remaining perpetuants, enumerated by
z
23
Z. Z
2 may be represented by the form
(1 T1+I)a(2'12+I)1,(lAl).(2g1+11)1,+... =(21'2+11 A1+1)1,
X1 and p2 each assuming all integer (including zero) values. For the case 0=0'=2, the condition is
e10,2r1T2(al +a2)(Q1 +r1) (01 +r2) =  Aa B1B2A1A2B = 0.
To represent the simplest perpetuant, of weight 7, we may take as base either A;IB1B2 orA1A2B , and since Al+B1=o the former is equivalent to AIM B2 and the latter to A2B1B,I; so that we have,
and
have been set forth above.
For the case 0=1, 0'=2, the condition is vlrlr2=A1B2=0; and the simplest perpetuant, derived directly from the product A1B2, is (I),,(2)1,(2I)); the remainder of those enumerated by
apparently, a choice of four products. 'A2BIB2 gives (22). (21)b—(221),(2)s, and AIA2B2,(22I).(2)6—(22),(21)5; these two merely differ in sign; and similarly A2B1B2 yields (2),(22I)b—(21),(22)b, and that due to AIA2B merely differs from it in sign. We will choose from the forms in such manner that the product of letters A is either a power of A1, or does not contain Ai; this rule leaves us with A2BjB2and A2B1B2; of these forms we will choose that one which in letters B is earliest in ascending dictionary order; this is A2B1B2, and our earliest perpetuant is
(22).(21)b—(221).(2)b,
and thence the general form enumerated by the generating function
27 is
(1 —z)(1 —z2)2
(2A2+2)a (2µ2+l1µ1+1)1,— (2A2+21). (2'12+11 X11) b + ...
\\ (2A2+2lµ1+1).(2'12+1)
For the case 8=1, 0'=3, the condition is e1r1r2r3(e5+rl)(al+r2)(al+7'3) =A1BafAiB2B3 =0. By the rules adopted we take A2B2B3, which gives
(12).(32)6 — (1),(321)6+ao(3212)b,
the simplest perpetuant of weight 7; and thence the general form enumerated by the generating function
zl
1—z.1—z2.1—z2'
viz:— (1 A3+2),(3LL3+12µ2+1)1,—...*a0(3µ3+I2µ2+11A1 )b,
For the case 0=2,8'=3, the condition is
0'10'2TIT2r3(al+0'2) (0'i +Tl) (0'1+7'2) (0'1+7'3) (0'2 1) (a2+7'2) (0'2+ Ts) X (Tl+r2) (T1+r3) (T2+r3) =0.
The calculation results in
—AlBsB2Bi+2AlBsB2B' —AzB3B2Bi+AaBsB1— 2A2BIB2B1 —A!,BZB2B8+AfB2BgB1+A2BgB2Bi +A2Ba B , 2A2B2B2 Bi +A2BsB1=0.
By the rules we select the product AZB3B2B8, giving the simplest perpetuant of weight 15, viz:
(24),(3212)b —(241).(321)s+(2412)o(32)b;
and thence the general form
(2A2+4)a(3µ3+42µ2+11µ1+2) b—... = (2A2+41'1+2)a(3µ3+12'12+1)b,
due to the generating function
(1—z) (1 —z2)2(1—z3) For the case 8=1, 0'=4, the condition is
0'1rIr2r3T4(0'l+7'i) (0'1 +T2) (0'1+T3) (0'1 +T4) II (Ts + I) = 0.
the calculation gives
End of Article: A4 (Z)
