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semi-variant forms established. Putting n equal to $\infty$, in a generating function obtained above, we find that the function, which enumerates the asyzygetic seminvariants of degree 0, is

${\frac {1}{1-x^{2}.1-x^{3}.1-x^{4}\dots \ 1-x^{\theta }}}$

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 ($3^{2}2),a(2^{4})$. 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

and the general form of perpetuant is $(4^{\kappa +1}3^{\lambda +1}2^{\mu })$.

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

In general when 0 is even and =2cf>, the condition is establishing the generators for 0 = 5, and 0 = 6, viz. :

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," Again, if 0 is uneven =20 + 1, the condition is it was noted that Stroh considers

where «r1 + <r2+...+ <r0 = O and “? = “?= ... = “*=«, symbolically, to be the fundamental form of seminvariant of degree 0 and weight w ; 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

If we expand the symbolic expression by the multinomial 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 shall obtain a sum of terms, such asw>! -L<?

which in real form is w !; and, if we express

in terms of A2, A3, ... and arrange the whole

as a linear function of products of A2, A3, ..., each coefficient will be a seminvariant, and the aggregate of the coefficients will give us the complete asyzygetic system of the given degree and weight. When the proper degree 0 is < w a factor a™ must be of course understood.

Ex. gr.

In general the coefficient, of any product A^A^A^..., will have, as coefficient, a seminvariant which, when expressed by paiti- tions, will have as leading partition (preceding in dictionary order all others) the partition (ttjTt^...). Now the symbolic expression of the seminvariant can be expanded by the binomial theorem so as to be exhibited as a sum of products of seminvariants,

of lower degrees if $\sigma _{1}\alpha _{1}+\sigma _{2}\alpha _{2}$ 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 form $\left(222\dots \right)A_{2}^{{\frac {1}{2}}w}$ 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 is ${\frac {z^{2}}{1-x^{2}}}$

For $\Theta =3,{(\sigma _{1}\alpha _{1}+\sigma _{2}\alpha _{2}+\sigma _{3}\alpha _{3})}^{w}$ ; the condition is clearly $\sigma _{1}\sigma _{2}\sigma _{3}=A_{3}=0$ , and since every seminvariant, of proper degree 3, is associated, as coefficient, with a product containing $\mathrm {A} _{3}$, all such are perpetuants.3 The general form is (3*2^) and the generating function <math>\frac{z^2}{1-z^2}<.math>

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

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

and the degree, in the quantities <r, is

2*+1+e*+>)+(V(+-+(T)

= 22^ - l = 2e_1 — 1.

> 2. The generating function 0is 1thus 3 (1 + cr^Xl + cr2|)...(l + <r0?) = 1 + A2f + A3f + ... + A0^. 2 “ -1 2 (1-z )(1-33)(1-24)---(1-20)

MacMahon to be a71"1 ay2 a™3 v irj ir2 , 4 k +2 1 o0-4 c 3 •••>

+1 ,0-2 c0-2 +2 ?-3 te-3+4 3 3 0-l ir1I 7T2! 7T3' ,2^ 1-1

...k2 being given any zero or positive integer values. 0 ^Simultaneous Seminvariants of two Binary Forms.—Taking the

two forms to be atfc+paix ^2+0(0 _ l)a2a3l x2 + .•■■-apX2,

b(fe + qbix ^x2 + q{q-)b2x x2 + ...+b(jc2,

every leading coefficient of a simultaneous covariant vanishes by

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

Observe obtain ic from any seminvariant one appertaining to a p- 1 ana

to

a q-l , and that suffix augmentation produces a, portion ol a

higher seminvariant, the degree in each case remaining unaltered.
Remark, too, that we are in association with non-unitary sym
metric functions of two systems of quantities which will De

S. I. — 39