S21 _ — ai Oaol — a2Ca01 — a13a30+a21
The formula actually gives the expression of () by means of separations of
(1Op014) ,
which is one of the partitions of q). This is the true standpoint from which the theorem should be regarded. It is but a particular case of a general theory of expressibility.
To invert the formula we may write
1 +alox +a03y +... +aP4xPy4 +...
=exp {(siox+soiy) s2ox2+2Slixy+so2y2) +...}, and thence derive the formula
()P+g1aD4
_ 1 (p1+q11)! R1 S (P2 +4; 1)! n2... (_)~,r1 n1 +~2
pl!gl! , p2!7ri!7r2I • 5711gt sp292•"' which expresses the elementary functions in terms of the single bipart functions. The similar theorem for n systems of quantities can be at once written down.
It will be shown later that every rational integral symmetric function is similarly expressible.
The Function hP4 .—As the definition of hp, we take
1 +n1ox +nmy b.. +n poxPy 4 + •
1
(1aix—thy) (1a2x—02y)...'
and now expanding the righthand side
rp, /J \\ rpz+421 ''' Plgl C2g2•••)
pi pz I `\
the summation being for all partitions of the biweight. Further writing
1 +hlox +holy +... +hP4xPy4+...
_ 1
1alox  aoly +... + ( )P+Qa P4xvy4 +.
we find that the effect of changing the signs of both x and y is merely to interchange the symbols a and h; hence in any relation connecting the quantities hP4 with the quantities aP4 we are at liberty to interchange the symbols a and h. By the exponential and multinomial theorems we obtain the results
2 1 (Eir)!
a a
(—) '1r1. 7r2.... Pigi p2g2 ••
and in this a and h are interchangeable.
(p+ql)! _ 1'i(E2r1)! .1
p! q! S55 ) 2ri! 7r2!...''PlgihP2g2'.. ;
)`S ,r
hD4= E (Pi 1) '2
! 'i S 2+ 2—1)! 1 *1 *g
l pr! q1! )( p2! q2!... S " 1ri! sr2!...SP1g15P2g2 ...
Differential Operations.—If, in the identity
(l +aix+Sly) (l +a2x+15'sy) ... (1+ a' x+$,.y)
= 1 +alox +aoly +a2ox2 +aaxy +a02y2 +... ,
we multiply each side by (1+µx+vy), the righthand side becomes
1+(aio+/3)x+(aol+v)y+...+(aP4+rta5_i,4+va5r4i)xvy4+... ; hence any rational integral function of the coefficients alo, aei, ... an, .•• say f(aio, am, ...) =f is converted into
exp(Eidlo+sdoi)f
where d10 = aP1,5 4 , dot = E ap,41 P4
The rule over exp will serve to denote that µdlo+Pdo1 is to be raised to the various powers symbolically as in Taylor's theorem.
Writing DPQ=p1q~d od 1,
exp(Eodlo+vdoi) = (1+AD10+vDo1+...+, 'v4DP4+...)f;
now, since the introduction of the new quantities o, v results in the addition to the function (p1g1p2g2p2q3.••) of the new terms
t2iv41(p2g2p3g3•••) +KP2v42(plgip3g3• ••) +113P43(piglp2g2•••) +••• , we find
DP1g1(p1g1p2gsp2q3...) = (p2g2psg3...) and thence
DP1g1DP242DPags... (plgip2g2p3g3...) = 1
while D„f=o unless the part r7 is involved in f. We may then state that DP. is an operation which obliterates one part pq when such part is present, but in the contrary case causes the function to
hP4 =
() Fq—1 h,4=
vanish. From the above Div is an operator of order pq, but it is convenient for some purposes to obtain its expression in the form of a number of terms, each of which denotes pq successive linear operations; to accomplish this write
dp4 =
~~IIl ar. an+r,s+.
and note the general result 1
exp (m,adio+moidoi+•..+mpgdpq+...)
=exp (MIodic +Moidol+.. +Mpgdpq+•••) ;
where the multiplications on the left and righthand, sides of the equation are symbolic and unsymbolic respectively, provided that mpg, bipg are quantities which satisfy the relation
exp (Mica+Mein+...+Mme'e+...)
=1 + micE + main +tna.,Onq+ ... ;
where E, n are undetermined algebraic quantities. In the present particular case putting m1o=µ, 'mot=v and mpg =o otherwise
M1oi;+Mein+...+Ma,ge'ng+... =log (1+µ>;+vn)
Mpg((p+a1)! pvg;
p,g!
and the result is thus
or
d
exp(ado +vdoi)
=exp(µdie+vdot 1
(µ2dfo+2avdu+v2do2)+...} =1+MD1o+pDo1+••.+µpv0Dpq+... ;
p+a1
( (pl+51_1)! Ri (p2+q21)! R2
Dpg
Ng1! p2!Q2!
... ir1!ir2!... Pan Tp2...,
the last written relation having, in regard to each term on the righthand side, to do with Evr successive linear operations. Recalling the formulae above which connect spg and apg, we see that dpq and Op, are in corelation with these quantities respectively, and may be said to be operations which correspond to the partitions, (pq), (iovolq) respectively. We might conjecture from this observation that every partition is in correspondence with some operation; this is found to be the case, and it has been shown (loc. cit. p. 493) that the operation
1 1 i dvigld,20... (multiplication symbolic)
corresponds to the partition (p,g1n1 p2g2"2•••)• The partitions being taken as denoting symmetric functions we have complete correspondence between the algebras of quantity and operation, and from any algebraic formula we can at once write down an operation formula. This fact is of extreme importance in the theory of algebraic forms, and is easily representable whatever be the number of the systems of quantities.
We may remark the particular result
(_)p+~'(P pIf 1)!dpgsp.2=Dpq(4 =1;
dpg causes every other single part function to vanish, and must cause any monomial function to vanish which does not comprise one of the partitions of the biweight pq amongst its parts.
Since
dpg=()"*'1(p+q 1)! d
p!q! dspg'
the solutions of the partial differential equation dpg=o are the single bipart forms, omitting spg, and we have seen that the solutions of D„g = o are those monomial functions in which the part pq is absent.
One more relation is easily obtained, viz.
d p dpqhlodp+1,qhold p,g+1+...+()*+.lh.dp+*,g+•+....
Ueber die Resultante eines Systemes mehrerer algebraischen Gleichungen," Vienna Transactions, t. iv. 1852; MacMahon, " Memoirs on a New Theory of Symmetric Functions," American
' Phil. Trans., 1890, P. 490.
Journal of Mathematics, Baltimore, Md. 1888189o; Memoir on Symmetric Functions of Roots of Systems of Equations," Phil. Trans. 189o.
A binary form of order n is a homogeneous polynomial of the nth degree in two variables. It may be written in the form
n ni n2 2
axl +bxi x2 +cx1 x2 +a.; or in the form
ax"+ll)bxn 1x2+2)cxnsx2+..., 1
which Cayley denotes by 1 z
(a, b, c, ...)(x1, x2)n
(1),(=)... being a notation for the successive binomial coefficients
n, in(nI),.... Other forms are
n n1 n2 2
ax +nbx x +n(nI)cx x +...,
1 1 2 1 2
the binomial coefficients (') being replaced by s!(7), and
n1 n2 2
axi +pbx x2+2 cxl x2+...,
the special convenience of which will appear later. For present purposes the form will be written
cox, +(,)aix, ix2tA2)aix22x2+...+a x2,
the notation adopted by German writers; the literal coefficients have a rule placed over them to distinguish them from umbral coefficients which are introduced almost at once. The coefficients ao, a1, ,...an, n+1 in number are arbitrary. If the form, sometimes termed a quantic, be equated to zero the n+1 coefficients are equivalent to but n, since one can be made unity by division and the equation is to be regarded as one for the determination of the ratio of the variables.
If the variables of the quantic f (x1, x2) be subjected to the linear transformation t
xl = anti + anti,
x2 = a21E1 + a2252,
f;2 being new variables replacing xi, x2 and the coefficients an, 902, a21, an, termed the coefficients of substitution (or of transformation), being constants, we arrive at a transformed quantic
—' R n _/ n—1 /R —' n2 f(E1, 2) =a E RI )a h +12)a h +...+a,.E2
where r= alla12 =aua12—a12a21; I a21a22
r is termed the determinant of substitution or modulus of transformation; we assure xi, x2 to be independents, so that r must differ from zero.
In the theory of forms we seek functions of the coefficients and variables of the original quantic which, save as to a power of the modulus of transformation, are equal to the like functions of the coefficients and variables of the transformed quantic. We may have such a function which does not involve the variables, viz.
F(ao, al, a2, n
...a) =r~F(ao, a1, a2,...an),
the function F(ao, a1, a2,...an) is then said to be an invariant of the quantic gull linear transformation. If, however, F involve as well the variables, viz.
F(ao, a1, a2,...;ti, Ei) =rAFCID, al, a2,...; xi, x2),
the function F(ac, a1, a2,... xi, x2) is said to be a covariant of the quantic. The expression " invariantive forms " includes both invariants and covariants, and frequently also other analogous forms which will be met with. Occasionally the word " invariants " includes covariants; when this is so it will be implied by the text. Invariantive forms will be found to be homogeneous functions alike of the coefficients and of the variables. Instead of a single quantic we may have several _
f (ao, a2...; ; x1, x2), 0(b_ o, b1, TT,— ; x1, x2), ...
which have different coefficients, the same variables, and are of the same or different degrees in the variables; we may transform them all by the same substitution, so that they become
f (ao, a1, a2,... ; t1, 6), ((ba, bl, b2 ... ; ti, E2),....
If then we find
F(ag, a1, a2,...bo, bt, b2, ; E1, 52),
s r 'F(ag, a1, e2,...bo, b1, b2, ; x1, Si),
and thence
µdio+vdoi _ (µ2d2c+2µvdu+v2do2) +••.
= log (1+A1)1o+vDe1+...+µvvgDpg+...).
From these formulae we derive two important relations, viz.
()s+g1(p+q—1)!d '()1,r1(Ea11)!Dr1 Dp2g2—..,
p!q! 9'2= ~ 2r1.2r2....
(_ E r1 ,r1 d7r2
o f 1 1 2 2 1 2
in the new variables which is of the same order as the original quantic; the new coefficients a , a , a ...a are linear functions
0 of the original coefficients, and also l linear functions of products, of the coefficients of substitution, of the nth degree. By solving the equations of transformation we obtain
rt1 = 0.22x1 a12x2, rf;2 =  a21x1+anx2,
the function F, on the right which multiplies r, is said to be a simultaneous invariant or covariant of the system of quantics. This notion is fundamental in the present theory because we will find that one of the most valuable artifices for finding invariants of a single quantic is first to find simultaneous invariants of several different quantics, and subsequently to make all the quantics identical. Moreover, instead of having one pair of variables x1, x2 we may have several pairs yl, y2; z1, z2;... in addition, and transform each pair to a new pair by substitutions, having the same coefficients an, 612, a21, a22 and arrive at functions of the original coefficients and variables (of one or more quantics) which possess the abovedefined invariant property. A particular quantic of the system may be of the same or different degrees in the pairs of variables which it involves, and these degrees may vary from quantic to quantic of the system. Such quantics have been termed by Cayley multipartite.
Symbolic Form.—Restricting consideration, for the present, to binary forms in a single pair of variables, we must introduce the symbolic form of Aronhold, Clebsch and Gordan; they write the form
n n n n1 nI n n n
alxl+a2x2)n =alxl+(l)al azxl x2+... I a2.x2=ay wherein a1, a2 are umbrae, such that
n1 n1 it
al. a1 a2,...ala2 , a2
are symbolical representations of the real coefficients ao, al,...
an1, an, and in general a~kaz is the symbol for ak.  If we restrict
ourselves to this set of symbols we can uniquely pass from a product of real coefficients to the symbolic representations of such product, but we cannot, uniquely, from the, symbols recover the real form. This is clear because we can write
  n1 n2 2 — 2n3 3
a1a2=a1 a2. a1 a2=a1 a2
while the same product of umbrae arises from
aoa3 =ai.ai 2a2 =a2n2 a2.
Hence it becomes necessary to have more than one set of umbrae, so that we may have more than one symbolical representation of the same real coefficients. We consider the quantic to have any
number of equivalent representations as=b:c:=.... So that ka2 = bl kb2  cl k
al c2 = ... = ak ; and if we wish to denote, by
umbrae, a product of coefficients of degree s we employ s sets of umbrae.
Ex. gr. We write alaz=ailaz•bl 2b2,
3
=a1 a2 n3 .b1 n3b28.c1n 3 8
a3e2,
and so on whenever we require to represent a product of real coefficients symbolically; we then have a onetoone correspondence between the products of real.coefficients and their symbolic forms. If we have a function of degree s in the coefficients, we may select any s sets of umbrae for use, and having made a selection we may when only one quantic is under consideration at any time permute the sets of umbrae in any manner without altering the real significance of the symbolism. Ex. gr. To express the function aoa2a;, which is the discriminant of the binary quadratic aoxi +2aix1x2+a2xa
=a~=bl, in a symbolic form we have
2(aoa2ai) =aca2+alai2a1 . a1 =aibi+az,bi 2a1a2b1bz = (alb2 a2b1)2.
Such an expression as aib2a2bi, which is
day abs acts abs
Oxi Ox2Ox2 Oxi'
is usually written (ab) for brevity; in the same notation the determinant, whose rows are al, a2, a3; b1, b2, b3; c1, c2, c3 respectively, is written (abc) and so on. It should be noticed that the real function denoted by (ab)2 is not the square of a real function denoted by (¢b). For a single quantic of the first order (ab) is the symbol of a function of the coefficients which vanishes identically; thus
(ab) =a1b2 a2bl=anal alao =0
and, indeed, from a remark made above we see that (ab) remains unchanged by interchange of a and b; but (ab) = (ba), and these two facts necessitate (ab) =o.
To find the effect of linear transformation on the symbolic form of quantic we will disuse the coefficients an, an, an, an, and employ X1, Al, X2, 132. For the substitution
XI = X151 +µ1S2, x2 = X251 +µ2S2,
of modulus a2 µ2 = (a1µ2X2111) =(aµ),
E
the quadratic form asx; +2alxlx2+a2x = y =f (x), becomes
Tot +2AlEli:2+Aata =At =~(>),where
Ao=aoXi+2a1X1X2+a2X2,
_A1 =aoXi 1+a1(t11p1±X2p1) +a2Xz/i2,
A2 =asp? +2a1µ1µ2 +a2µ 2 • We pass to the symbolic forms
ay = ((slx'+a2x2)2, A = (A161+A2 2)z, by writing for
an al, as the symbols a;, ala2, a;
and then Ao, A1, A2 ,, AT, A1A2, Aa
To = +2a1a2a1~2+az)2 = (ads'+a2Xz)2 =al, Al  = (silk +a2X2) (aiµ1+a2to2) =aAaµ, A2= (a1tL1+aisis)2=aµ;
so that

A~ +2aAaµEl6 +aµe = (axEl+aµ6)2;
whence Al, A2 become aA, aµ respectively and
~(f) = (aAf;l+aN,2)2.
The practical result of the transformation is to change the umbrae al, a2 into the umbrae
aA =a1)1+a2X2, aN, =a1µ1+a2tL2 •
Al = (al XI +a2X2) "1(a1µ1 +azµ2) = arcµ = Al 1 A2,
n k nk
Ak = (a1Xl+azaz)n k (a1µ1+azµ2)k =an aµk =A1 At
2,
so that the umbrae Al, A2 are ax, aµ respectively.
Theorem.—When the binary form
ay = (alxl+a2x2)n
A = (A151+A22)n
is transformed to
by the substitutions
x1=X1 1+111E2, x2 =A2f;1+µ2t2,
the umbrae Al, A2 are expressed in terms of the umbrae al, a2 by the formulae
Al =Xlal+X2a2, A2 =/Alai +µ2a2•
We gather that Al, A2 are transformed to al, a2 in such wise that the determinant of transformation reads by rows as the original determinant reads by columns, and that the modulus of the transformation is, as before, NO. For this reason the umbrae Al, A2 are said to be contragredient to xi, x2. If we solve the equations connecting the original and transformed unbrae we find
(Xµ) (— a2) = al (— A2) +µ1A1,
(X ') al = a2(  A2) +µ2A1,
and we find that, except for the factor (Aµ), a2 and +al are transformed to A2 and +Al by the same substitutions as xi and x2 are transformed to r and 2. For this reason the umbrae a2, al are said to be cogredient to xi and x2. We frequently meet with cogredient and contragedient quantities, and we have in general the following definitions:—(i) " If two equally numerous sets of quantities x, y, z,... x', y', z',... are such that whenever one set x, y, z,... is expressed in terms of new quantities X, Y, Z, ... the second set x', y', z', .. is expressed in terms of other new quantities X', Y', Z', , by the same scheme of linear substitution the two sets are said to be cogredient quantities." (2) " Two sets of quantities x, y, z, ...;
e, , ... are said to be contragredient when the linear substitutions for the first set are
x =>`1X+µ1Y+PIZ+...,
y = X2X+µ2Y+v2Z+...,
z = A3X +tz3Y +v3Z +...,
and these are associated with the following formulae appertaining to the second set,
_ X1E+X221+X3g+•••, H =µli+µz+1+µ3~+..., Z=st+v2n+Y3})+...,
wherein it should be noticed that new quantities are expressed in terms of the old, as regards the latter set, and not vice versa."
Ex. gr. The symbols dx, y dz, ... are contragredient with the variables x, y, z, ... for when
(x, y, z, ...) _ (a1, a1, v1, ...) (X, Y, Z, ...),
X2, /L2, V2, • I
X3, µ3, v3>
respectively.
By similarly transforming the binary n" form any we find
Ao = (a1X1+a2a2)n =as+
we find/
ad d ...> = (Xi, X2, a3, ...) ~dx , ay a , ...)
112, µ3, ... vi, V2, v3, •••
Observe the notation, which is that introduced by Cayley into the theory of matrices which he himself created.
Just as cogrediency leads to a theory of covariants, so contragrediency leads to a theory of contravariants. If u, a quantic in x, y, z, ..., be expressed in terms of new variables X, Y, Z ...; and
if n, r, . , be quantities contragredient to x, y, z, ...; there are
found to exist functions of t, n, ..., and of the coefficients in u, which need, at most, be multiplied by powers of the modulus to be made equal to the same functions of Z, H, Z, ... of the transformed coefficients of u; such functions are called contravariants of u. There also exist functions, which involve both sets of variables as well as the coefficients of u, possessing a like property; such have been termed mixed concomitants, and they, like contravariants, may appertain as well to a system of forms as to a single form.
As between the original and transformed quantic we have the umbral relations
Ai = Xiai+X2a2, A2 =Alai +µ2a2,
and for a second form
B, = aibi +)sbz: Bs = µibi +µ2b2•
The original forms are a:, b:, and we may regard them either as different forms or as equivalent representations of the same form. In other words, B, b may be regarded as different or alternative symbols to A, a. In either case
(AB) =AIBz—A,BI = (Xs)(ab);
and, from the definition, (ab) possesses the invariant property. We cannot, however, say that it is an invariant unless it is expressible in terms of the real coefficients. Since (ab) =aib3—a2b,, that this may be the case each form must be linear; and if the forms be different (ab) is an invariant (simultaneous) of the two forms, its real expression being agbi—aibo. This will be recognized as the resultant of the two linear forms. If the two linear forms be identical, the umbral sets al, a2; bi, b2 are alternative, are ultimately put equal to one another and (ab) vanishes. A single linear form has, in fact, no invariant. When either of the forms is of an order higher than the first (ab), as not being expressible in terms of the actual coefficients of the forms, is not an invariant and has no significance. Introducing now other sets of symbols C, D, ...; c, d, ... we may write
(AB)'(AC)~(BC)k... _ Ns) i+i+k+ ...( ab)ac) t(bc)k...,
so that the symbolic product
(ab)'(ac)i(bc)k...,
possesses the invariant property. If the forms be all linear and different, the function is an invariant, viz. the ith power of that appertaining to as and bx multiplied by the :eh power of that appertaining to as and cx multiplied by &c. If any two of the linear forms, say px, qx, be supposed identical, any symbolic expression involving the factor (pq) is zero. Notice, therefore, that the symbolic product (ab)`(ac)i(bc) ... may be always viewed as a simultaneous invariant of a number of different linear forms ax, bx, cx, .... In order that (ab)i(ac)i(bc)k... may be a simultaneous invariant of a number of
different forms ate', b':2, 42,..., where ni, n2, n3, ... may be the same or different, it is necessary that every product of umbrae which arises in the expansion of the symbolic product be of degree ni in ai, az ; in the case of 6,, b2 of degree n2; in the case of c2 of degree n3; and so on. For these only will the symbolic product be replaceable by a linear function of products of real coefficients. Hence the condition is
i+j +... =ni,
i+k+... = n2, j+k+••• =n3,
If the forms ate, cs,...be identical the symbols are alternative, and provided that the form does not vanish it denotes an invariant of the single form a:.
There may be a number of forms ai,bs,ci,... and we may suppose such identities between the symbols that on the whole only two, three, or more of the sets of umbrae are not equivalent; we will then obtain invariants of two, three, or more sets of binary forms. The symbolic expression of a covariant is equally simple, because we see at once that since Ai;, B , CE,... are equal to as, ba, cx, ... respectively, the linear forms ax, by, ca, ... possess the invariant property, and we may write
(AB)'(AC)'(BC)k...ABC... =
(~µ):+;+k+.. • lab)"(ac)' (bc)k...aPb°c'
x x x•••,
and assert that the symbolic product
(ab)' (ac)' (bc) k... amebic:...
possesses the invariant property. It is always an invariant or covariant appertaining to a number of different linear forms, and as before it may vanish if two such linear forms be identical. In general it will be simultaneous covariant of the different forms nl
bz 'P, c, ... if
i+j +...+P=ni, i+k+...+a=ns, j+k+...+T =n3,
It will also be a covariant if the symbolic product be factorizable into portions each of which satisfies these conditions. If the forms be identical the sets of symbols are ultimately equated, and the form, provided it does not vanish, is a covariant of the form a:.
The expression (ab)4 properly appertains to a quartic; for a quadratic it may also be written (ab)2 (cd)2, and would denote the square of the discriminant to a factor pres. For the quartic
(ab)4=(albs —a2bi)4=albs—4aia.sbibz+6aa2bib2 _ _ 4aia2bibs+alb'' = aaa4 4aiaa+6a' 4a,a3+aoa4 =2(aoa4—4a,aa+3a2),
one of the wellknown invariants of the quartic.
For the cubic (ab)2axbx is a covariant because each symbol a, b occurs three times; we can first of all find its real expression as a simultaneous covariant of two cubics, and then, by supposing the two cubics to merge into identity, find the expression of the quadratic covariant, of the single cubic, commonly known as the Hessian.
By simple multiplication
(albib2 2a?a2bib2+ala2ba)x?
+(aib2 ala2bib2—aia2b,b2+aabi)xixs +(aiasb2 2a,a2biba +a2bib2)xa ; and transforming to the real form,
(nobs — 2aibi+a2bo)xi +(aob3 —aib2 —ask +asbo)xixz + (aiba — 2a3bs+a3b,)x2,
the simultaneous covariant; and now, putting b=a, we obtain twice the Hessian
(aoa2—a?)x +(aoa3 —aias)xix2+(alas —a2)x8.
It will be shown later that all invariants, single or simultaneous, are expressible in terms of symbolic products. The degree of the covariant in the coefficients is equal to the number of different symbols a, b, c, ... that occur in the symbolic expression; the degree in the variables (i.e. the order of the covariant) is s+a+s+... and the weight' of the coefficient of the leading term xi+a+r+..• is equal
to i+j+k+.... It will be apparent that there are four numbers associated with a covariant, viz. the orders of the quantic and covariant, and the degree and weight of the leading coefficient; calling these n, e, 6, w respectively we can see that they are not independent integers, but that they are invariably connected by a certain relation nO—2w=e. For, if 4(ao,...xi, x2) be a covariant of order a appertaining to a quantic of order n,
(~+O r • • •S i53) =
(~/I)10~ (aor • • • X15, +FI1SZ r~251 +/+252) ;
we find that the left and righthand sides are of degrees nO and 2W+e respectively in Iii, µi, X2, µ2, and thence nO =2W+0.
Symbolic Identities.— For the purpose of manipulating symbolic expressions it is necessary to be in possession of certain simple identities which connect certain symbolic products. From the three equations
ax = a,xi+a,x2, bx = bixi+b2x2, Cy = cixl+czx2,
we find by eliminating xi and x2 the relation
ax(bc)+bx(ca)+cx(ab) =0 . (I.)
Introduce now new umbrae di, d2 and recall that +d2—di are cogredient with xi and x2. We may in any relation substitute for any pair of quantities any other cogredient pair so that writing +ds, —di for xi and x2, and noting that gx then becomes (gd), the abovewritten identity becomes
(ad)(bc)+(bd)(ca)+(cd)(ab) =0 . (II.)
Similarly in (I.), writing for ci, c2 the cogredient pair —y2,+yi, we obtain
axby—aybx=(ab)(xy). . ' . . (III.)
Again in (I.) transposing ax(bc) to the other side and squaring, we obtain
2(ac) (bc)axbx = (bc)2ai+(ac)2bx— (ab)2cs . (IV.)
and herein writing ds,—di for x2,
2(as)(be)(ad)(bd) =(be)2(ad)2+(ac)2(bd)2—(ab)2(cd)2. (V.)
As an illustration multiply (IV.) throughout by ax2bs2cs2 so that each term may denote a covariant of an nio.
2(as)(bc)a°'b2 c"1
x x x =(bc)2aibr+(ac)2a:2bzcx2—(ab)2as2hi72c .
' The weight of a tens., flood, ...d, is defined as being k,+2k2+... +nk,..
Each term on the righthand side may be shown by permutation of a, b, c to be the symbolical representation of the same covariant ; they are equivalent symbolic products, and we may accordingly write
2 (ac) (bc) a"ibn ~c 2 = (ab)2an2bn2cn
a relation which shows that the form on the left is the product of the two covariants
(ab)2ak 242 and c:.
The identities are, in particular, of service in reducing symbolic products to standard forms. Asymbolical expression may be always so transformed that the power of any determinant factor (ab) is even. For we may in any product interchange a and b without altering its signification; therefore
(ab)sm+141 = — (ab)2m+14
where 41 becomes 42 by the interchange, and hence
(ab)2m+1421 = 2(ab)2" (41 — 42) ;
and identity (I.) will always result in transforming 4142 so as to make it divisible by (ab).
Ex. gr.
(ab) (ac) bxc2, =  (ab) (bc)a.cz
=2(ab)c~{(ac)by(bc)as} =2(ab)2c;;
so that the covariant of the quadratic on the left is half the product of the quadratic itself and its only invariant. To obtain the corresponding theorem concerning the general form of even order we
multiply throughout by (ab)2si2cz°'2 and obtain (ab)2,n_t(ac) bscym1 =2 (ab)2mC2.m
Paying attention merely to the determinant factors there is no form with one factor since (ab) vanishes identically. For two factors the standard form is (ab)2 ; for three factors (ab)2(ac) ; for four factors (ab)4 and (ab)2(cd)2; for five factors (ab)'(ac) and (ab)2(ac)(de)2; for six factors (ab)6, (ab)2(bc)2(ca)2, and (ab)2(cd)2(e})2. It will be a useful exercise for the reader to interpret the corresponding covariants of the general quantic, to show that some of them are simple powers or products of other covariants of lower degrees and order.
The Polar Process.—The , polar of ay with regard to y is•
aniµay I+
,
i.e. A of the symbolic factors of the form are replaced by p others in which new variables yl, y2 replace the old variables x1, x2. ,The operation of taking the polar results in a symbolic product, and the repetition of the process in regard to new cogredient sets of variables results in symbolic forms. It is therefore an invariant .process. All the forms obtained are invariants in regard to linear transformations, in accordance with the same scheme of substitutions, of the several sets of variables.
An important associated operation is
02
a2
020y2—0x2aylr
which, operating upon any polar, causes it to vanish. Moreover, its operation upon any invariant form produces an invariant form.
Every symbolic product, involving several sets of cogredient variables, can be exhibited as a sum of terms, each of which is a polar multiplied by a product of powers of the determinant factors
(xy), (xz), (Yz),•..
Transvection.—We have seen that (ab) is a simultaneous invariant of the two different linear forms ay, b., and we observe that (ab) is equivalent to
where f = a=, 4 = bx.
If f = a: , 4 =b: be any two binary forms, we generalize by forming the function
(m k)! (nk)! (Of a4 _La4
ml n! \axlaxs ax2ax
This is called the kph transvectant of f over 4; it may be conveniently denoted by
(f, 4)k.
(am bn)k = (ab)ka'nkbnk
it is clear that the kth transvectant is a simultaneous covariant of the two forms.
It has been shown by Gordan that every symbolic product is expressible as a sum of transvectants.
If m>n there are n+1 transvectants corresponding to the values o, I, 2,... n of k; if k=o we have the product of the two forms, and for all values of k >n the transvectants vanish. In general we may have any two forms
4f = (41x1+42x2)v, 4i = (f'lxl+#2x2)4,
41, 4s, ¢2 being the umbrae, as usual, and for the km transvectant we have
(4i, 5~'y) k = (4+2)k4k4gk,
a simultaneous covariant of the two forms. We may suppose Of, 42 to be any two covariants appertaining to a system, and the process of transvection supplies a means of proceeding from them to other covariants.
The two forms a=, by, or 4z, 0g, may be identical; we then have the kth transvectant of a form over itself which may, or may not, vanish identically; and, in the latter case, is a covariant of the single form. It is obvious that, when k is uneven, the k°h transvectant of a form over itself does vanish. We have seen that transvection is equivalent to the performance of partial differential operations upon the two forms, but, practically, we may regard the
process as merely substituting (ab)k, (40)k for ayby, 4xk respectively in the symbolic product subjected to transvection. It is essentially an operation performed upon the product of •two forms. If, then, we require the transvectants of the two forms f +af , 4r+µ4', we take their product
f4+af'4+µf4'+xpf'4',
and the km transvectant is simply obtained by operating upon each term separately, viz.
(y y r 4)k+a(f', ~)k+ sc ( r ~')"+)(,/', 0n /k;
and, moreover, if we require to find the kth transvectant of one linear system of forms over another we have merely to multiply the two systems, and take the kth transvectant of the separate products.
The process of transvection is connected with the operations it; for ]
ilk (a'mbY/ nl = (ab)Oa. kamk bynk r
or f2k ((asbv)y_:= (f, 4)k;
so also is the polar process, for since
fkY =am.kb:,
k k
4y — by by,
if we take the km transvectant of fk, over 4y, regarding yl, y2 as the variables, j
(fk
4,:) k = (ab)k¢:mkbs k = l(J t, 4)k• ,
or the k°' transvectant of the km polars, in regard to y, is equal to the k' transvectant of the forms. Moreover, the km transvectant (ab)ka,, by k is derivable from the Ph polar of as, viz. a,kak, by substituting for yl, Y2 the cogredient quantities b2,b1, and multiplying by bzk:
First and Second Transvectants.—A few words must be said about the first two transvectants as they are of exceptional interest.
Since, if f =a,7, 4 =14,
(f, 4)1= I (Of 00Of 04) =(ab)az lbi1=J, mn axl axe Ox2 axl
the first transvectant differs but by a numerical factor from the Jacobian or functional determinant, of the two forms. We can find an expression for the first transvectant of (f, 4)1 over another form c9.
For
(m+n) (PO:, =nf•41y+mfy.0,
f,4y —fY•4 = (a:byayby) ay 'b:' = (xy) (f,4)1; (f,4)1 =f y4+m .Fn+n(xy)U,4)1.
Put m i for m, n 1 for n, and multiply through by (ab) ; then {(f =(ab)a'aybz't'm+ni2(xy)(f,4)2,
= (ab)a,'bzkby 4; I 2 (xy) (f,4)2•
Multiply byc T1 and for yl, y2 write 4,ci; then the righthand side becomes
(ab)(bc)az lbs2cri +m m+ ; ac:( f,O)',
of which the first term, writing cf =0, is
a, b,, cs2(ab)(bc)azcz
=  a,bykc ' 5 (be) 26z+(ab)2cx  (ac)2bs
_ 2 a,, (bc)2by s(((c: +cy (ab)2a b: _bs(ac)2aikc;k _23 (4,42)2•f+( ,42)2.0 —( ,4)2.4J ;
and, if (f,4)1=k'"+" 2
(f,4)1 ' 1.c9 1=k5+„2kyc'' 1• ,
A act, 00
axl Ox2 Ox2 ax1
Observing that
and
and this, on writing c2, c1 for y1, y2, becomes
(kc)k"'c: 1= # (f,m)1'# (1;
{(f,O)1,'F}1=1 m7;!.. 2 2 (O,P.f ;
and thence it appears that the first transvectant of (f, ¢)1 over 4' is always expressible by means of forms of lower degree in the coefficients wherever each of the forms f, ¢, 4' is of higher degree than the first in x1, Si.
The second transvectant of a form over itself is called the Hessian of the form. It is
(f ,f')2 = (ab)2a w 2bm2 = H="4 = H ;
unsymbolically it is a numerical multiple of the determinant
a2 a a2
It is also the first transvectant of the differaxf axe (axax2)
ential coefficients of the form with regard to the variables, viz. (( 11 For the quadratic it is the discriminant (ab)2 and for
\ax1 ' axe
the cubic the quadratic covariant (ab)2 a:bx.
In general for a form in n variables the Hessian is
a2f a2x a2f
ax! ax1ax2, ' axlax,, a2f ae a2f
ax1ax2 axa "'ax2axn
02f a2f a_
axlaxnax2ax,, " 04,
and there is a remarkable theorem which states that if H =o and n=2, 3, or 4 the original form can be exhibited as a form in t, 2, 3 variables respectively.
The Form f+a0.—An important method for the formation of covariants is connected with the form f+X4,, where f and ep are of the same order in the variables and a is an arbitrary constant. If the invariants and covariants of this composite quantic be formed we obtain functions of X such that the coefficients of the various powers of X are simultaneous invariants off and q,. In particular, when q, is a covariant of f, we obtain in this manner covariants of f.
The Partial Differential Equations.—It will be shown later that covariants may be studied by restricting attention to the leading
coefficient, viz. that affecting xi where a is the order of the covariant.
An important fact, discovered by Cayley, is that these coefficients, and also the complete covariants, satisfy certain partial differential equations which suffice to determine them, and to ascertain many of their properties. These equations can be arrived at in many ways; the method here given is due to Gordan. Xi, X2, /L2 being as usual the coefficients of substitution, let
x, +x2 =D xi— +x2a =D~K, t 2 AA au, u2
µtax,+1+2a a =Dµ,k, Ai— +P2— =DILA,
be linear operators. Then if j, J be the original and transformed forms of an invariant
w being the weight of the invariant. Operation upon J results as follows:
DaaJ=wJ; DA,,J=0;
D,AJ =0 jD,.wJ =wJ.
The first and fourth of these indicate that (aµ)'° is a homogeneous
function of a1, X2, and of µ1, u2 separately, and the second and third
arise from the fact that (Xµ) is caused to vanish by both DAµ and
Dµ,.
Since
we find that
(DA1,Ak) Ak =wJ; (DA Ak) Ak =0;
k k
1(D'`'`Ak)aa~Ak = 0; (DAk) A =wJ.
k k
According to the wellknown law for the changes of independent variables. Now DAAAk=(nk)Ak;DA Ak=kAk1;
so we obtain Dµ,Ak = (n —k)Ak}1;DµµAk = kAk; E (n—k)Aka =wJ;EkAk_,=0;
k k
(nk)Ak+, &= 0;EkAk A =wJ;
k kequations which are valid when X1, X2, ui, u2 have arbitrary values, and therefore when the values are such that J=j, Ak=ak. Hence
 aj a' nao= +(n1)a1_+(n2)a2 J +... _ aa° aa1 aa2
aj aj  aj
d°aa1+2a1Oa.2+3a2 a2+...=0,
 aj _ aj aj
nalaao I (n1)a2aa1+(n2)a3aa2+...=0,
 aj aj aj
a1aa1+2aeaa2+oa2aaa+... = wj,
the complete system of equations satisfied by an invariant. The fourth shows that every term of the invariant is of the same weight. Moreover, if we add the first to the fourth we obtain
 aj 2w._
akaak n OJ,
where 0 is the degree of the invariant; this shows, as we have before observed, that for an invariant
w=2n0.
The second and third are those upon the solution of which the theory of the invariant may be said to depend. An instantaneous deduction from the relation w=2n0 is that forms of uneven orders possess only invariants of even degree in the coefficients. The two operators
_— a a — a
= a0aa1 +2'aa2+... +na„laan
0 =nalaa0+(n 1)a2aa1+...+a" aa„_1
have been much studied by Sylvester, Hammond, Hilbert and Elliott (Elliott, Algebra of Quantics, ch. vi.). An important reference is "The Differential Equations satisfied by Concomitants of Quantics," by A. R. Forsyth, Proc. Lond. Math. Soc. vol. xix.
The Evectant Process.—If we have a symbolic product, which contains the symbol a only in determinant factors such as (ab), we may write x2,x1 for a1, a2, and thus obtain a product in which
(ab) is replaced by b5, (ac) by c,, and so on. In particular, when the product denotes an invariant we may transform each of the symbols a, b,...to x in succession, and take the sum of the resultant products; we thus obtain a covariant which is called the first evectant of the original invariant. The second evectant is obtained by similarly operating upon all the symbols remaining which only occur in determinant factors, and so on for the higher evectants.
Ex. gr. From (ac)2(bd)2(ad)(bc) we obtain (bd)2(bc)csda+(at)2(ad)cxdi
 (bd) 2(ad) azbx  (ac)2(bc) a ,bz
=4(bd)2(bc)cydz the first evectant;
and thence 4czdi the'second evectant; in fact the two evectants are to numerical factors pres, the cubic covariant Q, and the square of the original cubic.
If 0 be the degree of an invariant j
Oj = a=+aa' +... +a„a
aj n1 aj .aj
=alaao+al a2aa1+... +a2aa..
and, herein transforming from a to x, we obtain the first evectant
p,. ( — )kk,. kaj
z xx2 aak
k
Combinants.—An important class of invariants, of several binary forms of the same order, was discovered by Sylvester. The invariants in question are invariants qua linear transformation of the forms themselves as well as qua linear transformation of the variables.
If the forms be az, bi, ci,... the Aronhold process, given by the operation S as between any two of the forms, causes such an invariant to vanish. Thus it has annihilators of the forms
d d d
ae b°+a1ab1+a2db2+...
0ciao+bldal+b2da2+...
and Gordan, in fact, takes the satisfaction of these conditions as defining those invariants which Sylvester termed " combinants." The existence of such forms seems to have been brought toSylvester's
notice by observation of the fact that the resultant of az and bs must be a factor of the resultant of Xa +µb= and aas+µb' for a common factor of the first pair must be also a common factor
=H:(''2) =H. '
J =F(Ao,A1,.....), where kk =a aµ the results are equivalent to
of the second pair; so that the condition for the existence of such common factor must be the same in the two cases. A leading proposition states that, if an invariant of Xas and µb1 be considered as a form in the variables a and Al, _and an invariant of the latter be
taken, the result will be a combinant of as and b'. The idea can be
generalized so as to have regard to ternary and higher forms each of the same order and of the same number of variables.
For further information see Gordan, Vorlesungen fiber Invariantentheorie, Bd. ii. § 6 (Leipzig, 1887) ; E. B. Elliott, Algebra of Quantics, Art. 264 (Oxford, 1895).
Associated Forms.—A system of forms, such that every form appertaining to the binary form is expressible as a rational and integral function of the members of the system, is difficult to obtain. If, however, we specify that all forms are to be rational, but not necessarily integral functions, a new system of forms arises which is easily obtainable. A binary form of order n contains n independent constants, three of which by linear transformation can be given determinate values; the remaining n—3 coefficients, together with the determinant of transformation, give us n—2 parameters, and in consequence one relation must exist between any n—1 invariants of the form, and fixing upon n—2 invariants every other invariant is a rational function of its members. Similarly regarding xi, x2 as additional parameters, we see that every covariant is expressible as a rational function of n fixed covariants. We can so determine these n covariants that every other covariant is expressed in terms of them by a fraction whose denominator is a power of the binary form.
First observe that with f'2 = as = b= =...,f, =alas', f 2 = a2 a:1,
(ab)(af)—(bf)a,,.
and that thence every symbolic product is equal to a rational function of covariants in the form of a fraction whose denominator is a power of f. Making the substitution in any symbolic product the only determinant factors that present themselves in the numerator are of the form (af), (bf), (cf),...and every symbol a finally appears in the form.
= (af)kan,,
k
Y'k
!Gk has f as a factor, and may be written f. Ilk; for observing that
4'o=f. =f. uo; Y'1=0=f.u1; where uo=1, u1=o,
assume that ¢k = (af)ka:2 =f. uk =az.ukk0") Taking the first polar with regard to y
(n—k)(af)ka= k' +k(af)kia: (ab)(n—1)bs2by
n k,,2k1 n1 k(,.2)
=k(n2)azux uy+nay ayuy , and, writing f2 and —fi for yi and Y2,
(n — k)(af)k+ia, ki+k(n—1)(ab)(af)k1(bf)aa2b~2
= k (n 2) f. (uf) ny"2k1 Moreover the second term on the left contains
(af)`b: 2=~1 (af)k2by2—(bf)k2az 2}.
if k be uneven, and
(af)~1b: i= j(
C (af—(bf)1"a:l
if k be even; in either case the factor
(al) b. — (bf)= (ab)f,
(n—k)4Gk+1+M.f =k(n 2)f.(uf)
and 14+1 is seen to be of the form f .14+1. We may write therefore
uk = (af) ka:k
f
These forms, n in number, are called " associated forms " of f (" Schwesterformen," " formes associees ").
Every covariant is rationally expressible by means of the forms f, u2, since, as we have seen uo =1, u1=o. It is easy to find the relations
u2 = 2 (f f') 2
u3=((f ,f')2,f"')
u4=2(f,f')4•fi {(f,f')2}2,
and so on.
To exhibit any covariant as a function of u11, ui, u2,... take aY = (alyl+a2y2)"and transform it by the substitution
n.' f1Y1+f2Y2 = i; where fi=alans' ,fz=a2a: ,
x2Yi—xiy2 =n f =fixi+f2x2;thence
f . y, =xis+fzr1,'f• Y2 =x2k—fin, f.a,,=aki+(af)n,
j .a1,=u0 n+(2)u2 n2~2+(3)u32 3n~ l ...+u"n"•
Now a covariant of a: =f is obtained from the similar covariant
of av by writing therein x2, for y,, y2r and, since y,, y2 have been linearly transformed to E and n, it is merely necessary to form the covariants in respect of the form (ui+u2n)", and then division, by the proper power of f, gives the covariant in question as a function of fuo=1, u2, u3,...n,,.
Summary of Results.—We will now give a short account of the results to which the foregoing processes lead. Of any form ai there exists a finite number of invariants and covariants, in terms of which all other covariants are rational and integral functions (cf. Gordan,. Bd. ii. § 21). This finite number of forms is said to constitute the complete system. Of two or more binary forms there are also complete systems containing a finite number of forms. There are also algebraic systems, as above mentioned, involving fewer covariants which are such that all other covariants are rationally expressible in terms of them; but these smaller systems do not possess the same mathematical interest as those first mentioned.
The Binary Quadratic.—The complete system consists of the form
itself, as, and the discriminant, which is the second transvectant of the form upon itself, viz.: (f, f')2 = (ab)2; or, in real coefficients, 2(a0a2=ai). The first transvectant, (f,f')i=(ab) asbs,vanishes identically. Calling the discriminate D, the solution of the quadratic as =o is given by the formula
a 1
a,, a0 (acx,+a1x2—x2  doxi+alx2+x2' jI 3
If the form a' be written as the product of its linear factors p,,qz,
the discriminant takes the form —(pq)2. The vanishing of this
invariant is the condition for equal roots. The simultaneous system
of two quadratic forms as, ay, say f and 0, consists of six forms, viz.
the two quadratic forms f, 4.; the two discriminants (f, f')2,(0,0')2, and the first and second transvectants of f upon 0, (f, 0)1 and (f, 0)2, which may be written (aa)a,,a,, and (aa)2. These fundamental or ground forms are connected by the relation
21 (f,o)11 2=f2(d,,o')22fo(.f,o)2+0(.f,f )2
If the covariant (f,¢)' vanishes f and 0. are clearly proportional, and if the second transvectant of (f, 4,)i upon itself vanishes, f and 4. possess a common linear factor; and the condition is both necessary and sufficient. In this case (f, 0)1 is a perfect square, since its discriminant vanishes. If (f,4.)' be not a perfect square, and r,,, s,, be its linear factors, it is possible to express f and 4, in the canonical forms ai(r~)2+X2(s,,)2, t.i(r,,)2+µ2(s,,)2 respectively. In fact, if f and .0 have these forms, it is easy to verify that (f, 0)1= (aµ) (rs)r,,s,,. The fundamental system connected with n quadratic forms consists of (i.) the n forms themselves f ,
(ii.) the (2) functional determinants (fi,fk)', (iii.) the (2i) invariants (fi, fk)2, (iv.) the (3) forms (fi, (fk, f,,,))2, each such form remaining unaltered for any permutations of i, k, m. Between these forms various relations exist (cf. Gordan, § 134).
The Binary Cubic.—The complete system consists of
f =as , ( f,f')2 =(ab)2a,,bz =L1y, (f, A) =(ab)2(ca)bsc2 =Qzs, and
(0,0')2 = (ab)2(cd)2(ad) (bc) = R.
To prove that this system is complete we have to consider
(f,0)2, (0,0')1, (f,Q)i, (f,Q)2, (f,Q)', (A,Q)', (O,Q)2,
and each of these can be shown either to be zero or to be a rational integral function of f, 0 Qand R. These forms are connected by the relation
2Q2+A3+Rf2 =0.
The discriminant of f is equal to the discriminant of 0, and is therefore (0, 0')2=R; if it vanishes both f and 0 have two roots equal, . is a rational factor off and Q is a perfect cube; the cube root being equal, to a numerical factor pres, to the square root of
A. The Hessian A=A2 is such that (f,i)2=0, and if f is expressible in the form a(p,,)3+ i(gz)3, that is as the sum of two perfect cubes, we find that i1 must be equal to p.q for then
{a(p,,)3+µ(4%)3, psgs12=0.
Hence, if ps, q: be the linear factors of the Hessian Al, the cubic
can be put into the form X(p,,)3+u(q=)3 and immediately solved.
This method of solution fails when the discriminant R vanishes,
for then the Hessian has equal roots, as also the cubic f. The
Hessian in that case is a factor of f, and Q is the third power of
=fixi +f2x2, we find
and therefore
and
the linear factor which occurs to the second power in f. If, moreover, A vanishes identically f is a perfect cube.
The Binary Quartic.—The fundamental system consists of five forms a: =f; (f,f')2=(ab)2a:b2=As; (f,f')4=(ab)4=i; (f, A)1=
(a A) a:A: = (ab) 2 (cb) a2 bxcy = t ; (f,A) 4 = (aA) 4 = (ab) 2 (bc) 2 (ca) 2 = j, viz.
two invariants, two quartics and a sextic. They are connected by the relation
Remark.—Hermite has shown (Crelle, Bd. Iii.) that the substitution,
z ~ f, reduces x ~Xl Tx18xz to the form j
21z=2 f2A—A8—3jf3•
The discriminant, whose vanishing is the condition that f may possess two equal roots, has the expression j2—1—6.i8; it is nine times the discriminant of the cubic resolvent k32ik—3j, and has
also the expression 40, t')4. The quartic has four equal roots, that is to say, is a perfect fourth power, when the Hessian vanishes identically; and conversely. This can be verified by equating to
zero the five coefficients of the Hessian (ab) 2aib:. Gordan has also shown that the vanishing of the Hessian of the binary n`° is the necessary and sufficient condition to ensure the form being a perfect n°' power. The vanishing of the invariants i and j is the necessary and sufficient condition to ensure the quartic having three equal roots. On the one hand, assuming the quartic to have the form
4x ~x2, we find i= j = o, and on the other hand, assuming i= j = o, we find that the quartic must have the form asxi+4a1xix2 which proves the proposition. The quartic will have two pairs of equal roots, that
is, will be a perfect square, if it and its Hessian merely differ by a numerical factor. For it is easy to establish the formula (yx)2A4= 2f.f2y—2(fy)2 connecting the Hessian with the quartic and its first and second polars; now a, a root of f, is also a root of Ay, and con
sequently the first polar f1 —
u —y'axl+y2ax2 must also vanish for the
root a, and thence ixix and a must also vanish for the same root; which proves that a is a double root of f, and f therefore a perfect square. When f=6x1x2 it will be found that A=—f. The simplest form to which the quartic is in general reducible is f =xi+6mxix2+x2, involving one parameter m ; then A= = 2m (x: +x24) +2 (1—3m2) x2lx22 ;
i = 2 (i +3m2) ; j = 6m (1—m) 2 ; t = (1— 9m2) (xi —x2) (x2, +x2) x,x2. The sextic covariant t is seen to be factorizable into three quadratic factors 4)=xlx2, +y=xi+x22, 0=4—x2, which are such that the three mutual second transvectants vanish identically; they are for this
reason termed conjugate quadratic factors. It is on a consideration of these factors of t that Cayley bases his solution of the quartic equation. For, since 2t2=A3—2if2A—3j(—f)3, he compares the
righthand side with the cubic resolvent k22ia2k3jas, of f=0, and notices that they become identical on substituting A for k, and —f for a; hence, if k,, k2, k3 be the roots of the resolvent
212 = (A+k1f) (A+k2f)(A+k3f) ;
and now, if all the roots of f be different, so also are those of the resolvent, since the latter, and f, have practically the same discriminant; consequently each of the three factors, of 2t2, must be perfect squares and taking the square root
_ 1
t 24)•x•4);
and it can be shown that (A, x, ¢ are the three conjugate quadratic factors of t above mentioned. We have A+k1f=Ak2f=x2,
A+k3f=#G2, and Cayley shows that a root of the quartic can be
expressed in the determinant form
the remaining roots being obtained by varying
the signs which occur in the radicals (0v, xv, 4) . The transformation to the normal form reduces the quartic to a quadratic. The new variables
y,=0 are the linear factors of 4). If ¢=rx.sx, the
y2 =1 normal form of as, can be shown to be given by
(rs)'.ay = (ar) 4sz+6 (ar) 2 (as) 2rsss+ (as) ;
¢ is any one of the conjugate quadratic factors of t, so that, in determining rx, sx from sl A+klf =o, k, is any root of the resolvent. The transformation to the normal form, by the solution of a cubic and a quadratic, therefore, supplies a solution of the quartic. If (ap) is the modulus of the transformation by which a: is reduced to
is
the normal form, i becomes (Xu)4i, and j, (au)3j; hence j3 is absolutely unaltered by transformation, and is termed the absolute invariant. Since therefore 32 = 9 m2(11 3 m2)2 we have a cubic equation for determining m2 as a function of the absolute invariant.
1 j Oz
2i'J 2 1 1 j2 3
2+ T
3 2 a3
The Binary Quintic.—The complete system consists of 23 forms, of which the simplest are f = ay ; the Hessian H= (f, f) 2 (ab) 2a:b' ; the quadratic covariant 2= (Li) 5= (ab) 4axbx; and the nonic covariant T=(f,(f',f°)2)1=(f,H)'=(aH)a:H:=(ab)2(ca)azbycs; the
remaining 19 are expressible as transvectants of compounds of these four.
There are four invariants (i, i')2; (i3, H)6; (fz ia)m; (fl i7)14 four linear forms (f, i2)4; (f, i3)b; (i4, T)8 ; (is, T)9 three quadratic forms i; (H, i2)4; (H, is)'
three cubic forms (f,i)2; (f,i2)3; (is, T)6
two quartic forms (H, i)2; (H, i2)3.
three quintic forms f; (f,i)1; (is, T)4
two sextic forms H ; (H, i)1
one septic form (i, T)2
one nonic form T.
We will write the cubic covariant (f, i)2 = ', and then remark that the result, (f,j)8=o, can be readily established. The form j is completely defined by the relation (f, j)3 = o as no other covariant possesses this property.
Certain convariants of the quintic involve the same determinant factors as appeared in the system of the quartic; these are f, H, i, T and j, and are of special importance. Further, it is convenient to have before us two other quadratic covariants, viz. r = (j, j)2jxjx ; B = (ir)ixrx • four other linear covariants, viz. a=—(ji)2jx; =(ia)is; 7 =(ra)r:3=(rp)rx. Further, in the case of invariants, we write A= (i, i')2 and take three new forms B = (i, r)2; C = (r, r')2 ; R = (py). Hermite expresses the quintic in a formetype in which the constants are invariants and the variables linear covariants. If a, p be the linear forn}s, above defined, he raises the identity ax(ap)=ax(ap)—px(aa) to the fifth power (and in general to the power n) obtaining
(a$) sf = (a,5) sax—5 (a$) 4 (aa) az$x+... — (aa) bps;
and then expresses the coefficients, on the right, in terms of the fundamental invariants. On this principle the covariant j is expressible in the form
R2j =I +2B32a+4ACSa2+gC(3AB—4C)a8
when 3, a are the above defined linear forms. Hence, solving the cubic,
R2j = (5 —mia) (5 —m2a) (3 —m3a)
wherein ml m2, m3 are invariants.
Sylvester showed that the quintic might, in general, be expressed as the sum of three fifth powers, viz. in the canonical form f=k,(px)6 ±k2(gx)s+k3(rx)6. Now, evidently, the third transvectant of f, expressed in this form, with the cubic pxgxrx is zero, and hence from a property of the covariant j we must have j = pxgxrx ; showing that the linear forms involved are the linear factors of j. We may therefore write
f =kl(3—mia)b+k2(3—m2a)b+k3(5—m3a)6;
and we have merely to determine the constants k,, k2, lea. To determine them notice that R=(a3) and then
(f, ab) b = — Rb (k1+k2+k3) ,
(f, a43) 6 = — 5R5 (m1k1+m2k2+m3k2) , (f, a332) b= 10R5 (mlkl+mzk2+m3k3)
three equations for determining k1, k2, k3. This canonical form depends upon j having three unequal linear factors. When C vanishes j has the form j = psgx, and (f, j)3 = (ap)2(ag)as =o. Hence,
from the identity as(pq)=px(aq)—gx(ap), we obtain (pq)bf=(aq)bpy 5(ap)(aq)4pzgx—(ap)5q', the required canonical form. Now, when C=o, clearly (see ante) R2j=S2pwhere p=3+2Ba; and Gordan then proves the relation
6R4.f = B35+5B34p—4A2p6,
which is Bring's form of quintic at which we can always arrive, by linear transformation, whenever the invariant C vanishes.
Remark.—The invariant C is a numerical multiple of the resultant of the covariants i and j, and if C =o, p is the common factor of i and j.
The discriminant is the resultant of a and a and. of degree
8 in the coefficients; since it is a rational and integral function of the fundamental invariants it is expressible as a linear function of As and B; it is independent of C, and is therefore unaltered when C vanishes; we may therefore take f in the canonical form
6R4f B55 +5Bb4p— 4A2p6.
The two equations
ALGE'BRAIC'FORMS
a simultaneous invariant of the three forms, and now suppressing the dashes we obtain
6(abc +2fgh—aft bg2—ch2),
the expression in brackets being the .wellknown invariant of a:, the vanishing of which expresses the condition that the form may break up into two linear factors, or, geometrically, that the conic may represent two right lines. The complete system consists of the form itself and this invariant.
The ternary cubic has been investigated by Cayley, Aronhold, Hermite, Brioschi and Gordan. The principal reference is to Gordan (Math. Ann. i. 9o128, 1869, and vi. 436512, 1873). The complete covariant and contravariant system includes no fewer than 34 forms; from its complexity it is desirable to consider the cubic in a simple canonical form; that chosen by Cayley was ax3+by3+cz3+6dxyz (Amer. J. Math. iv. 116, 1881). Another form, associated with the theory of elliptic functions, has been considered by Dingeldey (Math. Ann. xxxi: 157176, 1888), viz. xy2—4z31g2x2y+g3x3, and also the special form axz2—4byb of the cuspidal cubic. An investigation, by nonsymbolic methods, is due to F. C. J. Mertens (Wien. Ber. xcv. 942991, 1887). Hesse showed independently that;the general ternary cubic can be reduced,
ab = 5B54+4B53p) =0, =5(B54—4Ap4) =0,
a
yield by elimination of 5 and p the discriminant
D=64B—A2.
The general equation of degree 5 cannot be solved algebraically, but the roots can be expressed by means of elliptic modular functions. For an algebraic solution the invariants must fulfil certain conditions. When R =o, and neither of the expressions AC —B2, 2AB 3C vanishes, the covariant a: is a linear factor of f ; but, when R =AC — B2 = 2AB 3C = o, a: also vanishes, and then f is the product of the
form ji and of the Hessian of jy. When as and the invariants B and C all vanish, either A or j must vanish; in the former case j is a perfect cube, its Hessian vanishing, and further f contains j as a factor; in the latter case, if px, ax be the linear factors of i, f can be expressed as (po)bf =cip;+c2vi; if both A and j vanish i also vanishes identically, and so also does f. If, however, the condition be the vanishing of i, f contains a linear factor to the fourth power.
The Binary Sextic.—The complete system consists of 26 forms, of which the simplest are f = as ; the Hessian H = (ab)2a;b6; the
quartic i= (ab)4a:bi ; the covariants 1= (ai)4ai ; T = (ab)2(cb)a. b:cy ; and the invariants A=(ab)b; B=(ii')4. There are
5 invariants: (a, b)4, (1,1')2, (f, 12)2, ((f, i) 14)2
6 of order 2: 1, (i,1)2, ,12)4, (i,12)3, (f, l3)2, ((f,i), 13)6;
5 of order 4: i, (f,l), (i,l) (f,12)3, ((f,i), 12)4;
5 of order 6: f, p = (ai)2a=is, (f, l), ((f,i), 1)2, (p, l) ;
3 of order 8: H, (f,i), (H,1) ;
1 of order 10: (H, i) ;
1 of order 12: T.
For a further discussion of the binary sextic see Gordan, loc. cit., €lebsch, loc. cit. The complete systems of the quintic and sextic were first obtained by Gordan in 1868 (Journ. f. Math. lxix. 323
r.). August von Gall in 188o obtained the complete system of the ary octavic (Math. Ann. xvii. 3152, 139152, 456); and, in 1888, that of the binary septimic, which proved to be much more complicated (Math. Ann. xxxi. 318336). Single binary forms of higher and finite order have not been studied with complete success, but the system of the binary form of infinite order has been completely determined by Sylvester, Cayley, MacMahon and Stroh, each of whom contributed to the theory.
As regards simultaneous binary forms, the system of two quadratics, and of any number of quadratics, is'alluded to above and has long been known. The system of the quadratic and cubic, consisting of 15 forms, and that of two cubics, consisting of 26 forms, were obtained by Salmon and Clebsch ; that of the cubic and quartic we owe to Sigmund Gundelfinger (Programm Stuttgart, 1869, 143); that of the quadratic and quintic to Winter (Programm Darmstadt, 1880) ; that of the quadratic and sextic to von Gall (Programm Lemgo, 1873) ; that of two quartics to Gordan (Math. Ann. ii. 227281, 1870); and to Eugenio Bertini (Batt. Giorn. xiv. 114, 1876; also Math. Ann. xi. 3041, 1877). The system of four forms, of which two are linear and two quadratic, has been investigated by Perrin (S. M. F. Bull. xv. 4561, 1887).
Ternary and Higher Forms.—The ternary form of order n is represented symbolically by
(aixi+a2x2+a3xi)" =as ;
and, as usual, b, c, d,... are alternative symbols, so that a"b"=c"=d"=...
.— s x .
To form an invariant or covariant we have merely to form a product of factors of two kinds, viz. determinant factors (abc), (abd), (bce), etc and other factors as, b., cx,... in such manner, that each of the symbols a, b, c,... occurs n times. Such a symbolic product, if it does not vanish identically, denotes an invariant or
a covariant, according as factors ax, b5, cx,... do not or do appear.
To obtain the real form we multiply out, and, in the result, substitute for the products of symbols the real coefficients which they denote.
For example, take the ternary quadratic
(aix,+a2x2+a3x3)2 =a2 x,
or in real form ax:+bxs+cx:+2fx2x3+2gx3xi+2hxix2• We can see that (abc)asbxcs is not a covariant, because it vanishes identically, the interchange of a and b changing its sign instead of leaving it unchanged; but (abc)2 is an invariant. If ax, b:, c= be different forms we obtain, after development of the squared determinant and conversion to the real form (employing single and double dashes to distinguish the real coefficients of ba and ex),
a(b'c"+b"c'—2f' ")+b(c'a"+c"a'—2g'g") +c(a'b"+a"b'—2h'h")+2"f(g'h"+g"h'—a'f"—a"f ) +2g(h'f"+h"f' —b'g"—b g')+2h(f'g"+f"g'—c'h"—c"h') ;
by linear transformation, to the form x3+y3+z3+6mxyz,
a form which involves 9 independent constants, as should be the case; it must, however, be remarked that the counting of constants is not a sure guide to the existence of a conjectured canonical form. Thus the ternary quartic is not, in general,' expressible as a sum of five 4th powers as the counting of constants might have led one to expect, a theorem due to Sylvester. Hesse's canonical form shows at once that there cannot be more than two independent invariants; for if there were three we could, by elimination of the modulus of transformation, obtain two functions of the coefficients equal to functions of m, and thus, by elimination of m, obtain 'a relation between the coefficients, showing them not to be independent, which is contrary to the hypothesis, .
The simplest invariant is S = (abc) (abd) (acd) (bcd) of degree 4, which for the canonical form of Hesse is m(I —m3) ; its vanishing indicates that the form is expressible as a sum of three cubes. The Hessian is symbolically (abc)2azbxcs = H3, and for the canonical form
(1{2m3)xyz—m2(x3+y2+z3). By the process of Aronhold we can form the invariant S for the cubic a=+XH!, and then the coefficient of a is the second invariant. T. Its symbolic expression, to a numerical factor pros, is
(Hbc) (Hbd) (Hcd) (bcd),
and it is clearly of degree 6.
One more covariant is requisite to make an algebraically complete set. This is of degree 8 in the coefficients, and degree 6 in the variables, and, for the canonical form, has the expression
9m6(x3+y3+z3)2 — (2m +5m4+20m7) (x3 +y'+z2)xyz
— (15m2+78mb 12m2)x2y2z2 + (1 +8m3)2(y3z3 +z3x3 +x3y3)•
Passing on to the ternary quartic we find that the number of ground forms is apparently very great. Gordan (Math. Ann. xvii. 21.7233), limiting himself to a particular case of the form, has determined 54 ground forms, and G. Maisano (Batt. G. xix. 198237, i88,) has determined all up to and including the 5th degree in the coefficients.
The system of two ternary quadratics consists of 20 forms; it has been investigated by Gordan (ClebschLindemann's Vorlesungen i. 288, also Math. Ann. xix. 529552) ; Perrin (S. M. F. Bull. xviii. 18o, 189o) ; Rosanes (Math. Ann. vi. 264) ; ' and Gerbaldi (Annali (2), xvii. 161196).
Ciamberlini has found a system of 127 forms appertaining to three ternary quadratics (Batt. G. xxiv. 141157).
A. R. Forsyth has discussed the algebraically complete sets of ground forms of ternary and quaternary forms (see Amer. J. xii. 16o, I15160, and Camb. Phil. Trans. xiv. 409466, 1889). Heproves, by means of the six linear partial differential equations satisfied by the Concomitants, that, if any concomitant be expanded in powers of x2, x3, the point variables—and of ui, u2, u3, the contragredient line variables—it is completely determinate if its leading coefficient be known. For the unipartite ternary quantic of order
n he finds that the fundamental system contains 2—(n+4)(n—1)
individuals. He successfully considers the systems of two and three simultaneous ternary quadratics. In Part III. of the Memoir he discusses biternary quantics, and in particular those which are lineolinear, quadratolinear, cubolinear, quadratoquadratic, cubo•cubic, and the. system of two lineolinear quantics. He shows that the system of the biternary n°m'° comprises
4(n+l)(n+2)(m+l)(m+2)—3 individuals.
Bibliographical references to ternary forms are given by Forsyth (Amer. J. xii. p. 16) and by Cayley (Amer. J. iv., 1881). Clebsch, in' 1872, in papers in Abh. d. K. Alead. d. U. zu Gottingen, t. xvii. and Math. Ann. t. v., established the important result that in the case of a form in n variables, the concomitants of the form, or. of a system of such forms, involve in the aggregate n—1 classes of
variables. 'For instance, those of a ternary form involve two classes which may be geometrically interpreted as point and line coordinates in a plane; those of a quaternary form involve three classes which may be geometrically interpreted as point, line and plane coordinates in space.
IV. ENUMERATING GENERATING FUNCTIONS
Professor Michael Roberts (Quart. Math. J. iv.) was the first to remark that the study of covariants may be reduced to the study of their leading coefficients, and that from any relations connecting the latter are immediately derivable the relations connecting the former. It has been shown above that a covariant, in general, satisfies four partial differential equations. Two of these show that the leading coefficient of any covariant is an isobaric and homogeneous function of the coefficients of the form; the remaining two may be regarded as operators which cause the vanishing of the covariant. These may be written, for the binary
n'
mkak_1 x2— =0; m(n—k)aa ik—xiad
=0;
or in the form
St—x2ax=0, O
where
d
d
=ao i+2aiaa }
d d d
O = naidao+ (n 1)a2aai +... +a,, .
Let a covariant of degree a in the variables, and of degree 0 in the coefficients (the weight of the leading coefficient being w and nO 2w =e), be
Cox: +ecixi 1x2+....
Operating with 0—xo we find SlCo=o; that is to say, Co satisfies
one of the two partial differential equations satisfied by an invariant. It is for this reason called a seminvariant, and every seminvariant is the leading coefficient of a covariant. The .whole theory of invariants of a binary form depends upon the solutions of the equation SZ=o. Before discussing these it is best to transform the binary form by substituting I !al, 2 !47,2, 3 ! a2,...n !an, for al, a2, aa...a,, respectively;
it then becomes
aexi+naixi 1x2 } n(n—1)a~xi~4 F...n!anx2; and 0 takes the simpler form
agcii +alaa2+a2aaa+'"+a~~i do
One advantage we have obtained is that, if we now write ao=o, and substitute a,_1 for a,, when s>o, we obtain
ao i+aid +a2 ...+a"'2danI
which is the form of ft for a binary
Hence by merely diminishing each suffix in a seminvariant by unity, we obtain another seminvariant of the same degree, and of weight w—o, appertaining to the (n—I)'. Also, if we increase each suffix in a seminvariant, we obtain terms, free from ao, of some seminvariant of degree 0 and weight w+o. Ex. gr. from the invariant al 2alai+2aoa* of the quartic the diminishing process yields ai2aea2, the leading coefficient of the Hessian of the cubic, and the increasing process leads to a; 2a2a*+2aia6 which only requires the additional term—2aoa6 to become a seminvariant of the sextic. A more important advantage, springing from the new form of 0, arises from the fact that if
x"—aixni+a2x"2—...(—)"an=(x—ai)(x— a2)...(x —an),
the sums of powers Eat, ma8, ma*, ...man all satisfy the equation 12=o. Hence, excluding ao, we may, in partition notation, write down the fundamental solutions of the equation, viz..
(2), (3), (4),...(n),
and say that with ao, we have an algebraically complete system. Every symmetric function denoted by partitions, not involving the figure unity (say a nonunitary symmetric function), which remains unchanged by any increase of n, is also a seminvariant, and we may take if we please another fundamental system, viz.
ao,(2), (3), (22), (32),...(21") or (321("a)).
Observe that, if we subject any symmetric function (pip2Ps...) to the diminishing process, it becomes aoi172 (P2P3...).
Next consider the solutions of 0=o which are of degree 0 and weight w. The general term in a solution involves the product av°aila= ...a*"wherein Mir=o, 157r, =w; the number of such products that may appear depends upon the number of partitions of w into o or fewer parts limited not to exceed n in magnitude. Let this number be denoted by (w; o, n). In order to obtain the seminvariants we would write down the (w; 0, n) terms each associated with
a literal coefficient; if we now operate with S2 we obtain a linear
function of (w—I; B, n) products, for the vanishing of which the
literal coefficients must satisfy (w—I; o, n) linear equations; hence
(w; 0, n)—(w—l; B, n) of these coefficients may be assumed arbi
trarily, and the number of linearly independent solutions of 12 =o, of
the given degree and weight, is precisely (w; B, n)—(w—I; 0, n).
This theory is due to Cayley; its validit _depends upon showing that
the (w—l; 0, n) linear equations satisfied by the literal coefficients
are independent; this has only recently been established by E. B.
Elliott. These seminvariants are said to form an asyzygetic system.
It is shown in the article on COMBINATORIAL ANALYSIS that (w; 0, n)
is the coefficient of aft' in the ascending expansion of the fraction
1
1—a. 1—az. 1—az2....1—aa"'
Hence (w; 0, n)—(w—l; 0, n) is given by the coefficient of aez' in the fraction
1—z
1—a.1—az. l —az2....I —az".'
the enumerating generating function of asyzygetic seminvariants. We may, by a wellknown theorem, write the result as a coefficient of en in the expansion of
1—zn+i .1—z"+2....1 —an+e
1—z2.1—zs....1—ze
and since this expression is unaltered by the interchange of n and o we prove Hermite's Law of Reciprocity, which states that the asyzygetic forms of degree B for the n44 are equinumerous with those of degree to for the o'°.
The degree of the covariant in the variables is e=no—2w; consequently we are only concerned with positive terms in the developments and (w, 0, n)—(w—I; o, n) will be negative unless no—2wo. It is convenient to enumerate the seminvariants of degree o and order a=no—2W by a generating function; so, in the first written
generating function for seminvariants, write a2 for_z and az" for a;
we obtain
1—z2
1—az". 1—azn2. 1—az"*....1—azn+*.1—azn+2.1—az"
in which we have to take the coefficient of aez"e21D, the expansion being in ascending powers of a. As we have to do only with that part of the expansion which involves positive powers of z, we must try to isolate that portion, sayAn(z). For n=2 we can prove that the complete function may be written
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