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THE ALGEBRA
OF
INVARIANTS
aonUon: C. J. CLAY and SONS,
CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,
AVE MARIA LANE.
ffilnsaofa: BO, WELLINGTON STREET.
Ifipjiji: F. A. BBOCKHAUS.
i^ffa gork: THE MACMILLAN COMPANY.
ISombne nnU CTalnitfa: MACMILLAN AND CO., Ltd.
[All Rights reserved.]
THE ALGEBEA
OF
INVARIANTS
BY
J. H. GRACE, M.A.
FELLOW OF PETEllIIOUSK
AND
A. YOUNG, M.A.
LECTUKER IN MATHEMATICS AT SELWYN COLLEGE, LATE SCHOLAK OF CLAKE COLLEGE
CAMBRIDGE:
AT THE UNIVERSITY PRESS.
1908
Cambrtligc :
PRINTED BY J. AND C. F. CLAY, /^'aK the ONlVE^Slit; PB'BJS. ".
/
Engineerinfj & Mathematical
Terences
Library
CO CD
CO
'StOl
P PREFACE.
ri^HE object of this book is to provide an English introduction
to the symbolical method in the theory of Invariants.
^ It was started as an attempt to meet the need expressed by
^;^ Elliott in the preface to The Algebiu of Qualities — 'a whole book
which shall present to the English reader in his own language
Nj a worthy exposition of the method of the great German masters
^ remains a desideratum.' Since then the need has been partly
^ met by the article 'Algebra' by MacMahon in the Supplement
v^ to the Encyclopcedia Britannica. The subject has been treated
from the commencement in order that readers unacquainted
with Elliott's treatise or any presentation of the elements may
4s be able to understand the argument. Such readers should bear
fM^ in mind that this treatise is only concerned with one part of
a very extensive subject. The modern theory of Partitions will
„ be found in the first part of the article by MacMahon mentioned
« above.
V The first six chapters — a great portion of which, we hope, will %- be found easy reading — may be said to lead step by step to N^j Gordan's wonderful proof of the finiteness of the system for a single binary form. The sixth chapter is, in fact, devoted to an exposition of Gordan's thii-d proof, but here, as throughout the book, we have allowed ourselves a free hand in dealing with the memoirs and treatises quoted. For example, we have made much use of Jordan's great memoirs on Invariants in proving Gordan's theorem: in a later chapter on Types of Co variants the development of Jordan's method has led us to some results which we believe
2609G3
VI PREFACE
to be important as well as novel, notably to an exact formula for the maximum order of an irreducible covariant of a system of binary forms.
The remainder of the book is mainly of geometrical interest : much space is devoted to Apolarity and Rational Curves, and the treatment of ternary forms is from the geometrical rather than the analytical point of view. The only complete system of ternary forms given is that for two Quadratics: it may be felt that more should have been said on this subject, but we think that with the methods known up to the present the treatment of ternary forms is too tedious for a text-book.
The number of references to Mathematical Journals etc. will perhaps be found unusually small : for this there is no need to apologise since the admirable Bericht ilber den gegenwdrtigen Stand der Invariantentheone* of Meyer gives references up to the last few years and in a more complete fashion than is desirable in a book which makes no pretensions to being exhaustive.
We wish to thank Dr H. F. Baker for help given to us in our early reading and Professor Forsyth for encouragement while writing. For reading of proof-sheets we are indebted to Mr J. E. Wright, B.A., of Trinity College, Mr P. W. Wood, B.A., of Emmanuel College, and in a still greater degree to the late Mr A. P. Thompson, B.A., of Pembroke College, whose Enthusiasm for Mathematics and research was most helpful and whose early death is deplored alike by his teachers and his fellow-workei*s. Our thanks are also due to the officials of the University Press for great help received during the course of printing.
J. H. GRACE. A. YOUNG.
* Jahresbcricht der Deutschen Mathematiker Vereinitfunrj, Vol, i., 1892. French translation byFehr; Gauthier-Villars, Paris, 1897. Italian translation by Vivanti; Pellerano, Naples, 1899. Article, Invariantentheorie in the Encyclopadie der iiMthcmatischen Winsenschaftcn.
August 18, 1903.
CONTENTS.
|
CHAP. I. |
Introduction |
PAQE 1 |
||||
|
II. |
The Fundamental Theorem |
21 |
||||
|
III. |
Transvectants .... |
36 |
||||
|
IV. |
Transvectants {continued) |
53 |
||||
|
V. |
Elementary Complete Systems |
85 |
||||
|
VI. |
Gordan's Theorem .... |
101 |
||||
|
VII. |
The Qcintic |
128 |
||||
|
VIII. |
Simultaneous Systems . |
158 |
||||
|
IX. |
Hilbert's Theorem |
169 |
||||
|
X. |
Geometry |
183 |
||||
|
XI. |
Apolarity and Rational Curves |
213 |
||||
|
XIT. |
Ternary Forms .... |
246 |
||||
|
XIII. |
Ternary Forms (coniinued) . |
274 |
||||
|
XIV. |
Apolarity (contimied) |
299 |
||||
|
XV. |
Types of Covariants |
319 |
||||
|
XVI. |
General Theorems on Quantics . |
339 |
||||
|
Appendix I. The Symbolical Notation |
365 |
|||||
|
)j |
II. Wronski's Theorem |
370 |
||||
|
>» |
III. Jordan's Lemma |
375 |
||||
|
Ii |
IV. Types of Covariants sidex |
. 378 . 381 |
4
CHAPTER I
INTRODUCTION. SYMBOLICAL NOTATION.
1. If in the expression
we write Xi = f^Xi + t/i X2 ,
W2= ^2X1 + 7)2X2, we obtain a new expression, viz.
AoXi' + 2A,X,X2 + A.,X2^ where ^0 = flo ^1^ + ^(h ^1 f 2 + ^2 ^2',
-4i = ao|i»7i + «! (|i7;2 + |2'7i) + a2^2'72, A2 = fto'/i^ + '^(hViV2 + (hvi- It is easy to verify the identity
A^A^-A^'^ (aoUo - ai") (li?/, - ^^Vif,
which shews that the function AqAs — A^^ of the coefficients of the transformed expression differs from the same function ao<*2 — ^i" of the coefficients of the original expression by a factor involving only the coefficients contained in the transformation.
2. In the present work we shall give an account of the theory and structure of functions of the coefficients possessing properties analogous to that described above ; but before pro- ceeding to generalities we shall give some further examples.
If we transform the two expressions
^01 •" ^Otiwju/2 "T" CLnOCi^ f
(^0 ^1 I ^(l\ X-yX^ T 0-2 X2 )
G. & Y. 1
2 . * THE ALGEBRA OF INVARIANTS [CH. I
in the same way as before, and they become
A;Xi^-\-2A,'X,X, + A,'X,\
then it is easy to verify the identity
AoA^' - 2A, A,' + Ao'A^ = {a^a^ -2a^a^'-\- a^a^) (^^^2 - ^^ViY-
Thus we have here a function of the coefficients of two ex- pressions such that the new value differs from the original value by a factor depending only on the transformation employed
3. As a third example, if the cubic expression
become A,X^' + ;Mx X;'X^ + ^A^X, Xi + A^Xi,
when we put a\ = l^jXi + t^iXj,
then we have
(^0^2 - A^) X^ + (^0^3 - ^1^2) ^1X3 + (^1^3 - A^) Z/
This identity indicates a property quite similar to that illus- trated in the two previous examples, but the function, which is unaltered except for the factor (^i?72 — ^2'7i)^ now involves the variables as well as the coefficients of the expression from which it is formed.
The result we have written down may be verified directly, but more easily as follows :
Denoting the original expression by f and the transformed expression by F we have to prove that
aAvaZa* \dx,dxj \dx,^dx^^ Kdx^dxJ]^^'^' ^'^'^^
dF__dF_dx^ dF_dx^ dXi dxi dXi dx^ dXi
'^'dx^'^^^dx^'
2-4] INTRODUCTION. SYMBOLICAL NOTATION 3
and in like manner
d^F d^F d^F d^F
d^F d^F d^F d^F
d^F ,d'F _ d^F ?'F
But these equations are exactly the same as those which express A^, A^, A2 in terms of Uo, a^, a^ (§ 1), hence
The expression ^^ ^7^ — 5~ra ) ^^ called the Hessian of/.
4. Let us now explain the phraseology in common use when dealing with questions such as arise in our subject.
Quantics. A rational integral homogeneous algebraic function of any number of variables Xj, x^, ... x^y is called a quantic.
The degree in the variables is called the order of the quantic,
and according as the number of variables is two, three, four
v.'e call the quantic binary, ternary, quaternary
Thus a binary quantic of order w is a rational integral homogeneous algebraic function of two variables which is of the nth degree in those variables.
Such a quantic might be written but we shall find it invariably more convenient to write it
i.e. with binomial coefficients prefixed to the various a's.
The former of these expressions is now commonly written
Vflo> ^1) *2> •••» (^nyj^\i '^2) >
and the latter {a^, a^, a^, ..., an\xi,x^^,
a very convenient notation introduced by Cayley.
1—2
4 THE ALGEBRA OF INVARIANTS [CH. I
The mere consideration of the transformation of the binary form
will be sufficient to convince the reader of the advantage of the introduction of binomial coefficients.
Passing now to the case of any number of variables, we call the quantic a p-ary ^-ic when it is homogeneous and of degree q in /) variables.
Thus the most general ternary quadratic is written and in general the ternary n-ic is written
where the summation is extended to all values of j9, q, r satisfying the equality
p + q + r = n.
It will be noticed that here we have prefixed multinomial coefficients to the as.
5. Linear Transformations. The equations
X.i = ^2^1 + '72>X^2
are said to constitute a linear transformation from the variables x^x^ to the variables X^X^ — it is of course implied that the coefficients on the right do not involve either set of variables.
The determinant
is called the determinant of the transformation.
If D vanishes it is evident that x^ and x^ are virtually identical, for their ratio is constant, and hence, as the variables are always supposed to be independent, we shall throughout only deal with transformations which have a non-vanishing determinant.
On solving for X^X^ we find
Xi = {7)^Xi-'r],X2)/D
*
Xi = {-^2Xi + ^iXi)II>
4-6] INTRODUCTION. SYMBOLICAL NOTATION 6
SO that the passage back from the new variables to the old is eflfected by a linear transformation. This is called the inverse of the original transformation ; it is evident at once that its
determinant is equal to ^ .
6. Let us now regard a linear transformation as an operator, which acting on a;^, x^ changes them to X^, X^, and let us consider the effect of two such operators acting successively.
If the coefficients of the first are and those of the second
y I ' . f- / /
?1 , ''71 , ?2 > '72 ,
then we have
a^i = ^i^i + »7i^2)
^2=^2^1 + »?2-X'2J '
-X'l=^l'^l' + ^l'Z2'| X2=^2X^ ■\-'q^X^\ '
and the effect of the two operators acting successively is to change from the variables cci, x^ to X/, X^.
Now on elimination of X^ , X^ we find
^i = (111/ + % la') X^ + (Iit;/ + -^iT?/) X^,
^2 = (I2I1' + ^7212') ^1 + (hvi + V2V2) X^'.
And accordingly we can pass directly from the original to the
final variables by means of a single linear transformation which
we shall call 2.
If we call the two preceding operators S and 8' we may write
X = SS'
and 2 is called the product or the resultant of S and S'.
It must be carefully noticed that the order of the factors S and S' is essential in considering their product. In our example we supposed that S acted first and then S'. If S' had acted first and then S we should have
X = S'S
and it is manifest that S and S' are not in general the same.
6 THE ALGEBRA OF INVARIANTS [CH. I
Since the resultant of two or any number of linear trans- formations is another such transformation, the whole set of linear transformations obtained by varying the coefficients is said to form a group — a continuous group because the coefficients ^ and 17 may be sxipposed to vary continuously.
The determinant of 2 is equal to the product of the determi- nants of S and 8', as follows from the multiplication theorem for determinants.
The product of a transformation and its inverse is a trans- formation which does not affect the variables, i.e. it is
which is called the identical operator. The determinant of this is unity, and, as we have pointed out, the product of the deter- minants of a transformation and its inverse is also unity.
7. The idea of a linear transformation admits of immediate extension to any number of variables x^, Xi,...Xp and now the transformation consists of n equations
Xr='^nX-^-\-^nX^+ ... + ^rpXj,, r=\, 1,...p.
The determinant D formed with the |'s for elements is called the determinant of the transformation, and inasmuch as when D vanishes there is a linear homogeneous relation between the x's, we exclude as before all transformations having a vanishing determinant.
If Z) 4= 0 we can solve for the X's in terms of the x& and, as can be easily seen, each X is a linear function of x^, x^, ... Xp, so that we have
Xr = Vn^i + Vr2^2 + • • • + Vrp^p>
a linear transformation which is the inverse of the preceding one.
As in the case of two variables, the resultant of two linear transformations S and T is a third linear transformation
and on examining the coefficients in S it will be seen at once by the multiplication theorem that the determinant of % is the product of the determinants of /S and T.
6-10] INTRODUCTION. SYMBOLICAL NOTATION 7
8. In the earlier portion of this work we shall deal almost entirely with binary forms, and although we shall be constantly considering linear transformations and their etfects, yet the fact that they form a group will not be explicitly used. Our only object, in introducing these elementary properties of groups, is to point out that the connection between invariants and groups is intimate and universal — in other words, that every group has its accompanying invariants and, conversely, every set of invariants belongs to a group.
9. Invariants of Binary Forms. If a binary form / be changed by a linear transformation into a new form F, and a function / of the coefficients of F be equal to the same function of the coefficients of / multiplied by a factor depending solely on the transformation, then I is called an invariant of the binary form/
Thus for example in § 1 the identity
(^0^2 - ^i') = (ao«2 - ai) (^iV2 - ^2Vi)" shews that a^a.^ — a-^ is an invariant of the binary quadratic
An exactly similar definition applies to a joint invariant of several binary forms, e.g.
a^a^ — ^a^tti + a^a^
is an invariant of the two binary forms
and adx^ + '2a-lx■^X2 + a^x^.
10. For the present we shall confine our attention to invariants which are rational integral functions of the coefficients. It is easy to see that there is no further loss of generality if we suppose the invariants to be homogeneous in each set of coefficients that they contain.
Thus for example if / be an invariant of a single binary form f which is not homogeneous in the coefficients a we can write / in the form /i + /a + . . . + /«,
where each element in this sum is homogeneous.
8 THE ALGEBRA OF INVARIANTS [CH. I
Now by definition we have
I{A) = MxI(a), and therefore
I,{A) + I,{A) + ... + Is(A) = M{I,(a)+I,(a) + ...+I,(a)}.
But the A's are linear functions of the as and M is independent of both, and therefore the only part on the left-hand side which is of the same degree as I^ (a) on the right-hand side is Ii{A);
.'. h{A) = Mh{a),
that is to say I^ is an invariant. Hence a non-homogeneous invariant is the sum of several homogeneous invariants.
This result can be at once extended to any number of binary forms.
As an example
ttoOs — a^ + a^a^ — 2a^a/ -I- a^ao is an invariant of the two binary quadratics
(tto, Oi, as 5^1, ^2? and (ao', a/, a^'^Xi, x^f, but it is the sum of two expressions
ttotta — tti",
and aoOa' — 2aia/ -I- a<iao
each of which is homogeneous in the two sets of coefficients.
11. Covariants of Binary Forms. If a binary form / is changed into a form i'^ by a linear transformation, and a function C of the coefficients of F and the new variables Xi, X^ be equal to the same function of the coefficients of /and the old variables ajj, a;, multiplied by a factor depending only on the transformation, then C is called a covariant of the binary form.
Thus from what we have seen
ay ay ( df Y
dxi^ dx^ \dxi dx^ j
is a covariant of the binary cubic/ and in fact of any binary form.
An exactly similar definition applies to a joint covariant of several binary forms — as an example the reader will have no difficulty in shewing that the Jacobian
§/;a5^_a/a0
dxi dx^ dx^ dxi
10-13] INTRODUCTION. SYMBOLICAL NOTATION 9
of any two forms / and ^ is a covariant of those forms, the multiplier being (^1% — |^2^i) the determinant of the transformation.
We shall confine our attention to covariants which are rational integral functions both of the coefficients and the variables, and, as in the case of invariants, there is no difficulty in seeing that there is no further loss of generality in supposing such covariants to be homogeneous in the variables and in each set of coefficients involved. In fact if a covariant be not homogeneous it is the sum of several parts each of which is a covariant and homogeneous.
12. Degree and Order of a Covariant. The degree of a covariant of a single form is its degree in the coefficients of that form — the order is the degree in the variables.
The covariant -~^ ^^ — f ^ ^ ) of a binary form of order n is
of degree two and order 2n — 4.
A covariant of several binary forms has a definite partial degree in each set of coefficients involved and the order is as before the degree in the variables.
The Jacobian of / and <^ is of degree one in the coefficients of each of the two forms, and its order is the sum of the orders of f and (f) diminished by two.
13. Symbolicar Notation. In our investigations we shall find it of the utmost value to write the binary quantic
ttoXi^ + naiXi^~^ 0)2+ ... +{ j a^a^i""'' a;/ + . . . + a^iPa" in the symbolical form
so that tti" = tto, tti"""^ a.2 = ttj , . . . a^"-'' a/ = ar,... ^^ = an.
This representation is startling at first sight, but consider how the use of it would introduce errors into calculation. They would arise because relations of the type
ana» = a
2?l— 2 ~ 2 /, 2
between the coefficients prevent our binary form from being a general one. Now in representing a function of the coefficients
10 THE ALGEBRA OF INVARIANTS [CH. I
symbolically we allow no symbol such as a to occur more than n times in any one term, so that the possibility of relations giving rise to
aotta = O]^
is entirely precluded. In fact to orbtain this relation there must be 2n a's multiplied together in the representation of the function aorta or al^ whereas, when we allow no more than n a's to occur in any one term, the (n+1) expressions
aA ai"-ia2,...ai"-'-a2'-, ...aa"
are independent quantities, i.e. with these restrictions on the use of our symbols the {n + 1) coefficients of the original quantic are not necessarily connected by any relation, and therefore the most general quantic can be represented in the form indicated.
Accordingly in addition to the symbol a we introduce a riumber of equivalent symbols /3, 7, ... so that
/= (aia^i + a^x^)'^ = (^^x^ + ^^x^Y = (y^x^ + y^x^Y =...
or as it will invariably be written
J — Clx—Px—'Yx •••
The symbolical equivalent of aorta is not
because here there are more than n a's multiplied together.
To represent rtort^ we must use two different symbols a, yS and then
aoaa = ai"/8i"-^A'> which is of course equivalent to
/9l''a,"-2a2^ whereas in the same symbols a,'' is represented by oii^~^okfii^~^^2'
In general to represent an expression of degree m in the coefficients, we have to use m different symbols of the type a, A 7. ••••
We have said that not more than n a's must be multiplied together in a given term — on the other hand if the expression has an actual as well as a symbolical significance not less than n of these symbols must occur together because only the expressions
ai", tti'^-^aa, ... Oa'' have an actual meaning.
13-14] INTRODUCTION. SYMBOLICAL NOTATION 11
14. A function of the coefficients can generally be represented symbolically in different ways as we have seen in the case of aoa, for example, which is equivalent to both
an^n-2^2 and ^^^a,-^-^.
There is one method of determining the symbolical repre- sentation which is very convenient because it often leads to the expression most suitable for our purpose.
Suppose, in fact, that P is a homogeneous function of the mih degree in Uq, a^, ... an, then
is only of degree m— 1 in a^, ai, ... an. If in Pi we replace each b by the corresponding a we obtain mP, as follows from Euler's Theorem relating to homogeneous functions.
In like manner if in
A i_Vfo~4-&— +b -)P
^9ai "" '^dajx'^dao ^ dai '" ^daj
we replace each c and each b by the corresponding a we get m (m — 1) P and P^ is of degree m — 2 in the a's.
Proceeding in this way we can find an expression Pm-i which is linear in each of m sets of symbols
a, b, c, ... k,
and which becomes equal to P x ml when each b, c, ... k is replaced by the corresponding a.
Now having formed the expression Pm-i we replace each a by the symbol a, each b by the symbol yS, each c by the symbol 7 and so on. Since the expression is linear in each set of letters, each symbol will occur exactly n times in every term, and then, regarding the symbols as referring to the same quantic, we have the required symbolical expression.
Thus for example
aoa, - a,^ = i (b, ^ + b, ^ + 6, Aj («„«^ _ a,%^^
= i (&o«2 + &2«o - 2ai6i)j=a
= i (AW + ^,W - 2aia,,8i A) a,-'^,^-'
12 THE ALGEBRA OF INVARIANTS [CH. I
and the convenience of this expression in terms of a, y8 will be abundantly evident in the sequel.
Ex. (i). For the binary quartic shew that
and flo «i «2 | = H«i/32-«2/3i)^Oiy2-^2yi)^(yia2-y2«i)*-
Ui a^ ct^ I
ttg CTg ^4 I
Ex. (ii). By the same method shew that for any binary form
Ex. (iii). Shew that for a binary form of odd order (a^^2 - cii&i)"' is zero and write down its value for a form of even order in terms of the coefficients.
15. Polar Forms. The expression (n-r)\f d d
y^^-^y-'
-Y/
where y*= aa;" = )Sa;'' = etc.
is a binary form of order n, is called the rth polar of/ with respect toy.
The operator ( t/i ^ — \-y^K-\, which is frequently written
l^'-a^. + ^'a^J-^
is said to be derived from / by polarizing r times with respect to y. The numerical factor ^^ -^ is only introduced for con- venience.
These polar forms admit of very simple representation in our symbols, for
and so on.
Hence the rth polar of / with respect to y is (n — r)!
w!
w (71 - 1) . . . (n - r + 1) (x^-''tLy\
that is OiJ^-^'aJ.
14-16] INTRODUCTION. SYMBOLICAL NOTATION 13
The differential coefficients of f with respect to the variables are particular cases of polar forms.
For if 2/1 = 1, 2/2 = 0, the rth polar is
ay
aa;/ and if y-^ = 0, 3/2 = 1, the rth polar is
9^2""' In general we have
aP+«/" n !
J^ — /y n-p—qn Pn 9
^x^-dxS {n-p-q)\ "" ' '•
The form ^ ^ [y^] (•^5~)/^^ called a mixed polar
with respect to y and 2 ; its symbolical expression is
a^n-p-q^a^q^
16. Effect of a Linear Transformation. If we write ^^1 = ^1^1 4- 771X2
^2 = ^2X1 + Vi^z, then Ux becomes
or ajZi + a^Xg,
and hence the binary form a^^ becomes
(afZi + a,Z2)"
or ai" Xi" + n aj'*-^ a, Xi"-^ X^+...+ a^X^.
Accordingly in the transformed expression the coefficient of Xj** is found by replacing x by ^ in the original form, and the coefficients of Xi'*~^X2, X-^~'^X} ... are found by polarizing the
coefficient of X-^ with respect to i) once, twice Of course
suitable numerical multipliers must be introduced.
The reader will easily illustrate this result by reference to the transformation of a binary quadratic in § 1.
14
THE ALGEBRA OF INVARIANTS
[CH. I
|
= |
«!, |
"2 |
X ^1, |
^. |
|
^1, |
A |
Vi, |
V2 |
17. The form a^^ = ^x^ = 7a;" • . . becomes on transformation (a^X, + a,X,r = (^f Z, + /3,X,r = (y^X, + y.X.y =....
Now we have
All + Af2, /3l»7l + A'?2
= («! A - otsA) (^i'72 - fa'/i), a result of fundamental importance.
We shall denote the expression (ai/Sa — tto/Si), which we call a symbolical determinantal factor, by (a/3), so that (ay3) = — (/9a), and the above relation may be written
af/g,-a,;8i = (a/3)(^^).
To illustrate these remarks let us prove that
do 0^4 — 4aia3+ Soa^
is an invariant of the binary quartic
We have
(00^4 - ^cb\(h + 302=*) = :| f 26 5- j (ttotti - 4aia3 4- ^a^)h=a = i (a/8)*.
Thus, if the coefficients of the new form be denoted by capital letters as usual, we have
(^0^4 - 4^i^3 + 3^2^) = i (aj A - a, A)*
as follows from the symbolical expression given above.
But since a^ ^, - a,ySf = (a/9) (^17),
A^A^ - 4^i^3 + 3^2^ = {^tjY (aoa^ - ^a^a^ + ^(h%
which shews that a^ai— 4iaiaz + Za^ is an invariant and that the multiplying factor is the fourth power of the determinant of transformation.
18. Symbolical expressions representing Invariants.
If the symbolical equivalent of an expression /, homogeneous and of degree % in the coefficients of the binary form
be an aggregate of terms each of which is a product of factors of the type (ay6), then / is an invariant of the quantic.
17-19] INTRODUCTION. SYMBOLICAL NOTATION 15
For let I = ^T, where T is the product of w factors of the type (ajS), then the total degree of T in the symbols is 2w and it is also n x i, for there must be i sets of symbols, each set occurring to degree n\ therefore ni=2w, so that w is the same for every term in the aggregate representing /.
If /' be the same function of the coefficients of the transformed expression, then /' = ^T'
where T' is found from T by replacing a^ by a^ , a.2 by a, and so on.
But since (a^/S, - a,/3f) = (a/S) {^t})
it follows at once that T = {^y T,
and therefore I' = {^7))^ I since w is the .same for every terra.
Hence / is an invariant.
Exactly the same result is true for any number of binary forms if we suppose that / is homogeneous in each set of coefficients, for it is easily seen that the number of determinantal factors must be the same in every term, it being in fact
when Til, n^, ... are the orders of the forms and i^, i^, ... the respective degrees of / in the coefficients of the forms.
The rest of the proof then depends only on the fact that, whatever a and /3 are, we have
Thus / is an invariant and the multiplying factor is now
19. This simple theorem enables us to construct as many invariants as we please — we have only to write down a product of factors {a^) and take care that the symbol a occurs in n of these factors where n is the order of the form to which the symbol a belongs. If this condition be not satisfied the invariant property still holds but the expression has only a symbolical meaning. On the other hand, if every symbol occur to the right degree but the expression be not reducible to the form above, it is an actual function of the coefficients which is not an invariant.
16 THE ALGEBRA OF INVARIANTS [CH. I
As an example we have an invariant of the second degree (fliS)" for a binary form of order n. This vanishes identically when n is odd, as can be seen by expressing it in terms of the' coefficients ; or thus, since a, /3 are equivalent symbols
and hence (a^)" = (-l)"(ayS)",
giving the result at once.
Again, for the binary cubic we have the invariant
and for the binary quartic the invariants
(a/3)^ (a^n^yYiyay, {a^y(ay)(^8){y8y.
In every case it will be observed that the multiplying factor is a power of {^v)-
As an example of invariants of several binary forms we may mention (ayS)", an invariant of the two different binary forms fla:" and /9a;". For quadratics this is the well-known invariant of §2.
Again (a/3) is an invariant of the two linear forms a^ and ^^ and in this case a, /8 are actual coefficients as well as symbols. Then (ayS) (ay) is an invariant of the quadratic Ox^ and the linear forms /3a,, y^.
20. Covariants. A similar method exists for constructing covariants.
Commencing with an example let us prove that the Hessian
^_ayay /ayy
dxi^ dxi \dxidxj is a covariant of the binary form /= a^^ = /3a;" = . . . Since H is of the second degree in the coefficients
^ V daJ Ida;,' dx^^ \dx,dxj j b=a
\dxi^ dx^ "dx^ dx^ dx^dx^ dx^dxj 6=a where /' = {\h^ . . . bn$XiX^y\
19-20] INTRODUCTION. SYMBOLICAL NOTATION 17
Replacing the as by as and the 6's by /3's as usual, we have
H=^n'' (n - 1)2 {ai^a^'^-^ye^^yga.'^-a
+ a.^a,-^ A^^^-2 - 2a,ct,a^"-^ A/82/3«,"-='j
as can be immediately verified by expressing this in terms of the coefficients.
The transformed quantic is
and the corresponding expression derived from this is
(a^/3, - a,/3|y^ (ajZ, + a^X^y-'i^^X, + ^,X,f-'
which shews that the expression is a covariant and that the multiplying factor is (^■j/)^
In general, if an expression C, of degree i in the coefficients of/ and of order m in the variables, can be symbolically repre- sented as an aggregate of terms, each of which is the product of a number of factors of the type (a/S) and a number of the type a-t, then G is a covariant oif.
In fact let C = SF, where F is such a product.
The number of factors with suffix x in F must be m, the order of G, and if w be the number of the type (a/8) we have
2w + m = ni,
for each of these represents the degree of G in the symbols. Hence lo is the same for every term.
If C" be the corresponding expression derived from the transformed quantic, then
G' = 2F',
where F' is derived from F by replacing «! by a^, a^ by a, and so on, and «« by {a^X^ + a^Xg).
Thus since (a^^, - a,/3f) = (a/3) (^t;)
and a^ = a^Xj + a^Xj
we have F = {^r}Y F,
.-. C" = (|^)«'C,
that is to say C is a covariant.
G, & Y. 2
18 THE ALGEBRA OF INVARIANTS [CH. I
Exactly the same method applies to a covariant of any number of binary forms, but now the symbols a, /S, ... may refer to different forms and, of course, a symbol such as a. must occur in the symbolical expression to the requisite degree.
We can thus easily construct any number of covariants of one or more forms, e.g. for a binary form of order n
is a covariant for any integral value of r, but it vanishes when r is odd.
Again, if <x^, ^^ are two different quantics,
is a covariant ; if r = 1 it is the Jacobian.
As further examples we have the covariants
(a/3)» {&if ina)- ct,;S,7„ (ay8) {^i) (7a) (aS) (^g) (7S) a^'^i^A^
of the binary quintic
a.'' = y8.' = 7^» = a/=...
As an exercise the reader may prove that the last one vanishes identically.
21. We have seen how useful the symbolical methods are in constructing invariants and covariants. In the next chapter we shall prove that they constitute an ideal calculus when we shew that every invariant and covariant can be represented as a sum of symbolical products of factors of the types (a/S) and a^. Meanwhile anticipating this result we shall indicate the methods of trans- forming symbolical expressions. These depend on two principles :
(i) Interchange of equivalent symbols,
(ii) Identities in symbolical expressions.
According to (i) if a symbolical expression have an actual meaning and contain two equivalent symbols then its value is not altered by interchanging those symbols. We have already used this method in proving that the invariant
(ayS)" of the quantic a^^ = /3^"
20-22] INTRODUCTION. SYMBOLICAL NOTATION 19
vanishes when n is odd. As another easy example we have
(ay8)(/37)(7a) = 0 for the quadratic a^^ = jSx^ = yj', or for the two different quadratics a^^ = ^^^ and yx^. More generally the co variant
(ayS) (M (7^) ax''^^x''-V-'
is alw^ays zero unless the three forms a^^, /3a;" and jx^ are all different.
22. Fundamental Identities. We have identically
(Moix + (ya)^x + {ci^)yx==0 (I).
as can easily be verified.
From this identity many others may be deduced. For example, replacing cc^ by B2 and x^ by — Si we have
(/37)(aS) + (7a)(;S8) + (a/3)(7S) = 0 (II),
a result useful in transforming invariants. Again from (i)
(M ^x = (^a) Ix - (7«) ^a; and hence by squaring
2 (otyS) {ay) ^xlx = (a/3)-^ Ix' + {ayf ^x' - {^yf a,^ • • .(HI). As identities less generally used we may mention
{^yf ax' + (7a)' ^x' + {oi^r Ix' = 3 (M (7a) (^'/S) aa=/3a=7*
()S7)'«.^ + (7«)^/Sa.^ + («;Sy7*'
= 2 {(a;8)^ (a7)-^ /3,.^7,,^ + {^yf {^af yx^ai + (7«)' (7/^)^ a*^/3«=1.
Ex. (i). For the quadratic
(a^) (ay) /3,y, = ^ {(a^)^ y^ + (ay)^ ^,2 - {^yf a,^}
since the symbols a, /3, y are equivalent.
2—2
20 THE ALGEBRA OF INVARIANTS [CH. I
Ex. (ii). If fi = a^=^^ and fi — ^x=^x be two different quadratics, to express the square of the Jacobian J={aa)axa'x in terms of/ and /'.
We have J^={aa') a^a'^ {m') ^x^'x
or since (|3/3') a'^={^a') ^'^ - O'a') ^^
J^={aa') i^a') a,^,^J^- {aa') O'a') a,^'^ . /3,2
= i3',2 i {{aa'f ^,2 + Oa')2 a,^ - {a^f a',^}
-^,m(aa'r^,^ + {P'ayaJ-{a^ya'J^} by (III),
or if (a/3)2=/,i, (aa')2 = (a0')^=...=/,2, {a'^f = l^^,
we have 2 J2 =/2 {/j^^/-^ + I^J^ _ /^^/J
~/i{m2/2 + A2/1 ~ ■'12/2}
Ex. (iii). Prove that for the binary quartic
(a^)2(ay)2^,2y^2=.|_^.(„^)4 (o^) (ay) a,2/3,3y^33=^y. („^)2 „^2^^2.
CHAPTER 11.
THE FUNDAMENTAL THEOREM.
23. It will be remarked that in every example of in- variants and covariants, discussed in the preceding chapter, the symbolical expression for such a function involved only factors of the types (a/3) and a^;, and further that the multiplier alluded to in the definition was always a power of the determinant of the transformation. We are now going to establish the general truth of these properties.
As a matter of history, we may observe that the original definition of an invariant stated that the multiplier was of the form mentioned ; but following the logical, rather than the historical order, we shall first prove that the multiplier must be a power of the determinant and then proceed to prove the proposition relating to the symbolical forms for invariants and covariants.
24. Suppose that / is an invariant or covariant of a single binary form f — after what has been said, § 10, we may assume that I is homogeneous in the coefficients of/.
Let the linear transformation
change / into /' and let /' be formed from /' in the same way that / is formed from f\ then, by definition,
/'=i^(^i,77i, f 2,772) x/
and we have to shew that F is simply a power of (f 1772 — ^2''7i)-
22 THE ALGEBRA OF INVARIANTS [CH. II
Now let a second transformation
Xi ^= q\Xi + 7Ji X^ X^ ^ 52 '^\ "■" '72 ''^2
change /' into /", and let /" be formed in the same way from f", so that
Hence we have
/" = ^(ri. '71. ^2, 1.) X F{^,\ 7;/, ^/, ,;;) X /.
But we can pass from the variables x^, x.^ to the variables x", x^' by the single transformation
Xi = (fi^i' + 171I2') a;," + (^i77i' + T/iW) a^a"
a^2 = (^2^1' + '72I2') ^1" + (^2';i' + '72'?2') ^2" ;
therefore
Consequently F must satisfy the functional equation
= F(^„ 71,, ^2, %) X F{^^, V. r;, ^72'). The solution of this equation is not difficult. In the first place we remark that since ^1 = 1, 7/1 = 0, ^2 = 0, % = 1 gives the identical transformation,
F{\,0,i),l) = \.
Again putting ^1 = k, 7/1 = 0, I2 = 0, '72 = ": each new coefficient is equal to the corresponding original coefficient multiplied by the same power of k, in this case the multiplier is clearly a power
of K, i.e.
F{k, 0, 0, /«;) = «:'■. Since
F{^u Vi,h, V2) X F(k, 0, 0, k) = F(k^„ KVi, K^2, 'CV2), we have
F(k^i, /CT/i, K^2, icVt) = K-^'Fi^i, 77i, ^2. '72),
therefore F is homogeneous and of degree r in the four variables 1^1. ^71. 1^21 Vi-
Finally let us choose ^Z, t;/, I2'. %' so that
^i|i' + 771^2' = 1 > |i»7i' + »7i'72' = 0. ^2^^/ + ^72^2' = 0, |2'7i' + '72^2' = 1.
^B 94-_9fi1
24-26] THE FUNDAMENTAL THEOREM 23
which relations give
{D = ^j772 - ^27/3),
then we have
Fi^,, Vl, ^2, V2) X i^(^/, %', f/, 772') = i^(l, 0. 0, 1)
Consequently since ^ is homogeneous and of degree r
F (^1 , %, I2, '^a) X i' (772, -vi, - ?2, f ) = ^.
But inasmuch as D is obviously irreducible — i.e. it cannot be resolved into factors — and F is clearly an integral function, this equation shews at once that both
-^(^1. Viy ^2, V2) and F(r}2, - r)^, - ^2, ^1)
are integral powers of D.
Hence the theorem is established.
25. Assuming the truth of the proposition just proved, the proof of the fundamental theorem that invariants and covariants can be completely represented by factors of the types (a/8) and a^ is very simple in principle. The actual work requires two lemmas of great importance in the present subject, and we shall give them separately. Tbey are both concerned with properties of the differential operator
26. Lemma I. If n be a positive integer
In fact ^— (x^y^ - x^y^Y = ny^ {x^y^ - x<^iT~^,
+ ?? (n - 1 ) (^13/2 - iio.jy^y-^'X{y^
24 THE ALGEBRA OF INVARIANTS [CH. II
Similarly
92 ^— - {x^y^ - x^,y =-n {x^y^ - x.y^f-^ ^n{n-V) (x^y^ - x^y.f-^x^y, .
Consequently
n (^i2/2 - x^y^Y ={n(n-l) + 2n} {x,y., - x^y.y-' = n{n+l) {x^y., - x^yif-\ which establishes the lemma.
If we operate again with H we find
fi2 (^x^y^ - x^y^Y = n{n + \){n-\)n{xiy^- x^y^f-'' and in general
= (n + l)n'{n-iy ...{n-r + 2)^ {n-r + 1) {oc^y^-x^y-,Y-^ or n*- {xyY = (n + 1) /i^ {n-iy ... {n-r + 2)^ {n-r + 1) {xyy^. Finally
n« (a^i 2/2 - ^22/1)" = (n + 1) (w n^'
a constant which is not zero — for our immediate purpose this is the important result, and it can be at once verified by expanding
\dx,dy~ dx,dyj ' ^^^2/2-^2^1) by the Binomial Theorem.
27. Lemma II. If the operator fl be applied r times to the product of m factors of the type a^. by n factors of the type fiy, then each term in the resulting expression contains r deter- minantal factors {ol^), (m — r) factors Ox and {n — r) factors ^y.
To ensure perfect generality we consider
where P = Oa,*" aa,<''' . . . a^;*'"'
and Q = /3/)^/'.../3,<»',
the a's and the ^'s being all different.
H 26-28] Wm Now
26-28] THE FUNDAMENTAL THEOREM 25
where the summation extends so that r takes all the values 1, 2, ... m and s takes all the values 1, 2, ... n.
Hence on subtraction
P Q
nP.Q = SK)/3'«0.,,,^„.
which establishes the lemma for r = 1.
But since the operator CI has no effect on a factor of the type (a"''yS<**) the theorem holds for r = 2 ; in fact
and performing the operation on the right we have the result.
Proceeding in this way we see that at each step a new factor of the type (a/S) appears in each term while one factor of each of the types ax and /3y disappears — this completely establishes our lemma.
Ex. (i). Prove that Q'' a '" 6/ = , , , , ^ (ahY a^-"" 6 "-^
Ex. (ii). With the notation of the text prove that Q'' F. Q contains every term of type there written r ! times and that the number of different terms is
, '~^, J — ". , — , . (Use induction.)
(m — r)! («.-r) ! r! ^ '
28. Fundamental Theorem. Suppose now that P(ao, ai, ... a„) is an invariant of the binary form
(tto, ai, ... a„][^i, x.^"" = a^" = ^^« = ..., then after the linear transformation
^2 = ^2^1 + •J72^!
26 THE ALGEBRA OF INVARIANTS [CH. II
we have seen that a^ becomes aj and a^ becomes a, ; so that if the new form be
we have Ar = aj^^^a/.
By definition
F(Ao, A„ ... Ar,) = (I1172 - hv^yFia,, a„ ... a„) ;
accordingly if the A's are replaced by their symbolical expressions F becomes the sum of a number of terms, say XP . Q, wliere P contains only factors of the type a^, and Q only those of the type a,. As the degree in ^ and rj must be w we infer that there are just w factors in P and w in Q.
If we operate on both sides with D,'^, P . Q becomes the sum of a number of terms each of which is the product of w factors of the type (ayS), and the result on the right-hand side is a numerical multiple of J^(ao«i ... ««)•
Hence we have expressed F(ao, ai, ... an) in the symbolical form peculiar to invariants.
The proof as given applies to invariants of one binary form ; it is the same, word for word, for any number of binary forms, for the left-hand side is still of equal degree in f and rj, and on the right-hand side we have the determinant {^r)) occurring to a power equal to this degree. Hence operating as above the required symbolical expression is obtained.
29. The proof for covariants is of the same nature as that for invariants, although a little more care is required in the manipulation of the symbols ; after what has been said on in- variants we may confine our attention to covariants of a single form.
Suppose that F{ao, a^, ... an, oci, x^
is a homogeneous covariant of order m of
(tto, Oi, . . . ttn^iCi, X^y = Oa,^ = /8/ = etC.
Then using the same notation as before F{Ao, Au ... An, X^, X^) = (^iV^-^-iViyPicio, ai. ••• ««, ^i, ^2).
28-30] THE FUNDAMENTAL THEOREM 27
If the A's are replaced by their symbolical expressions we get
where P involves only factors of the type a^ and Q only those of the type a^.
But on solution we have
fl'72-t2'7l Cl»72-?2%
Now for convenience we shall replace x^ by Wg s-nd a;^ by — Mi, so that
X, = ,, "% ,, Z, = -
Wf
(|l'72 - ^2'7l) ' (|l';2 - f2'7l) *
Substituting these values and multiplying up by (^rj)^ we obtain the identity
2 (- 1)"*= P . Qu^'^iu^'"^ = (^t; )«-+"» i^.
We may write the left-hand side XP' . Q' where P' only contains factors with suffix f and moreover exactly (w + m) factors, and Q' contains {w + m) factors with suffix 77.
Accordingly after operating with 11^+*" each term will involve {lu + m) determinantal factors, and the right-hand side will be a numerical multiple of F. Now of the {w-\- m) factors, there are w of the type (ayS) and m of the type (aw), for u must occur to degree m in the final as well as in the original expression.
But {au) is — OLx ', hence replacing the it's by the xb throughout we have the symbolical expression for F.
30. Since we have proved that all invariants and covariants of one or more binary forms can be completely represented by products of factors of the types (a/8) and ax, and further that every expression which can be so represented is an invariant or covariant — provided it possesses an actual significance, — it follows at once that all properties of invariants and covariants are implicitly contained in the symbolical representation and can be deduced therefrom.
28 THE ALGEBRA OF INVARIANTS [CH. II
31. Let us examine somewhat more closely the constitution of invariants of a single binary form
/= (tto, tti, . . . an^xi, Xof = OL,'' = ^x" = 7*" etc.
Suppose that an invariant / is an aggregate of products of factors (a/S) such that in every term there are w factors, then inasmuch as each symbol occurs n times in / we must have
ni = 2w,
where i is the number of different symbols.
Now the weight of a,, is r by definition and its symbolical equivalent is ai^'^a/; hence the weight of any product of the as is the sum of the weights of the factors and is therefore equal to the total degree to which the letters ««, ySg, 72, ... occur in the symbolical equivalent.
In the case of an invariant such as / each term in the symbolical expression when multiplied out is the product of w symbols with suffix 1 by w symbols with suffix 2 ; hence the weight is w. Further, the multiplying power of the determinant for I is also w.
Consider next a covariant of degree i and order m. It is an aggregate of terms, each of which is the product of the same number (say p) of factors of the type (a/8), by the same number (say q) of factors of the type Ox.
We deduce at once the relations
q = m, 2p + q — ni,
for each member of the latter equation represents the total degree of the covariant in the symbols a, /S, 7,
Thus m = q = ni— 2p.
32. The leading coefficient of the covariant is called a seminvariant — it is found at once from the covariant by putting
cci = l, X2 = 0
and therefore represented symbolically it is an aggregate of products of p factors of the type (a/3) by q factors of the type ai.
31-33] THE FUNDAMENTAL THEOREM 29
The weight of this seminvariant is accordingly p, and hence we infer that if w be the weight of a seminvai'iaut and i its degree the order of the corresponding covariant is ni— 'ho.
Thus for example in connection with the cubic
we have the covariant {a^ya^^x- The seminvariant is
i.e. its weight is 2, as we should have inferred from the number of factors (olIS) in the covariant.
The order of the covariant is 2 and here we have
i =2, n — 2, 7n= 2, w = 2,
so that m = ni — 2w.
In like manner if the leading coefficient of a joint covariant of two quantics of orders Jij and n^ be of degrees i^, 4 in the respective coefficients and of total weight w in these coefficients conjointly, then the order of the covariant is
riiii + 7?2t2 — 2w.
The reader will readily establish this theorem and extend it to the case of any number of quantics by using the symbolical notation. On putting 771 = 0 we get a relation connecting the degrees and weight of an invariant.
33. Deduction of a covariant trova. its leading coeJBEicient.
As we have seen, each term in the symbolical expression for the seminvariant must be the product of w factors of the type (a/9) by m factors of the type ai-
Now suppose that in the seminvariant we replace ai by CLg, A by ^x, etc., and leave unaltered a., /S.,, etc., then (a/3) becomes
(oiiTi + aoX^ A - (ySia?i + ^^x.^ tta = (a/3) on^ ;
hence the seminvariant S is clearly changed into x^^ multiplied by the corresponding covariant — e.g. in the cubic, (a/3)"^ai/3i becomes xi^ X {oi^foix^x- We have thus a simple means of passing from the leading coefficient to the covariant. A similar result for
30 THE ALGEBRA OF INVARIANTS [CH. II
invariants may be obtained by taking the particular case m = 0 ; here the leading coefficient is of course the invariant itself.
Let there be an identical rational algebraic relation among a number of seminvariants S^, S2, ... Sr and let G^, G^, ... Gy be the corresponding covariants.
If the relation be and Wp be the weight oi Sp, then the sum
must be the same for every term — hence if we put the left-hand side of the relation into symbols and then change a^ into ax, /9i into ^x as above, we have
or 2(7j'*'C/»...a/' = 0,
i.e. the covariants are connected by the same relation as the seminvariants.
34. Again when we replace Ui by Ox and leave a^ unaltered we replace the coefficient a^ by
ax^-^OLr=
_(n-r)l 9^ n! dx/
Hence except for a multiplier, which is a power of x, a covariant is the same function of
/ 1 _§t 1 dY {n-ry. ay i ay
as the corresponding seminvariant is of
©0 , Ctj , £12 > • • • ^r > • • • ^n •
Ex, (i). If in a seminvariant of weight w we replace ag by a^, /Sg by ^x, etc., and leave aj , ^^ . . . unaltered, then the result is the seminvariant multiplied by ^Tg".
What is the corresponding transformation of the actual coefl&cients ?
Ex. (ii). Find the result of replacing a^ by a* , og by a,, , ^i by /Sf , /Sj by ^T, , etc., in a seminvariant, and give the corresponding transformation of the coefl&cients.
33-35] THE FUNDAMENTAL THEOREM 31
Ex. (iii). Extend all the above results to the case of two or more binary forms.
Ex. (iv). Prove that if in an invariant of a single binary form a^ be
(n-r)\ d^f replaced by 7-^ ^—^ and so on, the result is the invariant multiplied by
(n-r)\ d^f x^. State the result of replacing a^ by -^ j-^' ir-^ and extend the argument
to any number of binary forms.
35. Alternative proof of the Fundamental Theorem — The Aronhold Operator.
We shall now give another proof of the theorem of § 28 in which the original argument of Clebsch will be followed.
Let 0 be a covariant of a form
/=(ao, tti, ... an][a?i, x^""
which is homogeneons in both the coefficients and the variables, and of degree i in the former.
If F = {A„A,,...An\X,,X^-
be the transformed quantic, we have
</)(J.o, ^1, ... ^„) = /A</> (tto, ai, ... a„)
where /x. depends only on the transformation.
Now if (&o. ^1, ••• ^n][«i, ^2)"
be a second form which transforms into
then (tto + >^^o, tti + X6i, ... an + ^h^x^, x^"^
transforms into
(^0 + Xi?o, 4i + \5i, . . . ^« + X5„][Z„ X^f . Therefore
</)(A + ^5o, ^i + X-Bi, ... ^„ + \5,,)
= yu,^ (tto + X6o, tti + X6i, . . . a» + A.6„), hence expanding by Taylor's Theorem and equating coefficients of X we have
32 THE ALGEBRA OF INVARIANTS [CH. II
therefore the expression on the right is a joint covariant of the two forms ; in other words the property of invariance is not affected by an operator like
. da) "" V ° 9^0 ^9^1 * * " " da J '
Hence, proceeding exactly as in § 14, we can construct a covariant of i different quantics which is linear in the coefficients of each, and which becomes a numerical multiple of (f> when we replace each of the i quantics hyf.
The operator (&^) is called the Aronhold operator; its
importance lies in the fact that it enables us to construct simul- taneous invariants or covariants of several binary forms of the same order when any invariants or covariants are known for a simple form.
Thus, for example, since aoa-i—a^^ is an invariant of the quadratic
(I/Q OC-y "7~ ^ (X\ w/j 00)^ "^ ttg U/2
the expression aJ)^-{-aJ)Q— 2ai6i is a simultaneous invariant of the two quadratics
(tto, ai, a^x^, x^^Y and (6o, h, h^x^, x^\
The construction of other illustrations will present no difficulty.
36. It has been proved in § 14 that any covariant of degree i can be symbolically represented as a function </> of degree n in the coeflBcients of each of n different linear forms
Oi, )3z> Jxy ••• '■>
since ^ is a covariant of the original quantic it is unaltered by any linear transformation, except for a factor which depends only on the transformation, hence also it is a covariant of the i linear forms.
By further use of the Aronhold operator we can now find a covariant of ni different linear forms
a^W, a;,(2), ... ax(")
)3:c(i), 0A ... /3x(») etc.
linear in the coefl&cients of each form, and such that it becomes a numerical multiple of the original covariant when each of the symbols
a(»), a(2), ... a(»)
is replaced by a, each of the symbols
^(1), /3(2), ... /3(») by /3, and so on.
35-37] THE FUNDAMENTAL THEOREM 33
We need therefore only consider linear covariants of linear forms in the sequel ; we shall prove that every covariant of a system of linear forms is a rational integral function of invariants of the type (a/3) and covariants of the type ax- Once this is established the general theorem follows immediately.
37. System of Linear Forms. First consider a single linear form
and let (p be any invariant or covariant.
If we use the linear transformation given by
where b is any constant, then the new linear form is Xi and we have the equation
where fi depends only on the transformation, i.e. only on a^, ag, b.
Now let (f) = -\}rQai^ + ylria{'~^a2+ ...+'^r<i2'
where the \|/'s do not depend on oj, ag but contain only x^, x^.
Then *=%J/ + ^I'i^i'-M2+...+*r^2''
where ^^ is the same function of JSTj, Xg as x//-^ is of o^j, ^g > ^od further -4^ = 1, ^2 = 0 since the transformed form is X^.
Hence *o = M^ = /^(>^oai'' + ^iai''"^a2+---+^ra2'") (I)-
Now ^0 depends only on X^, JTg, therefore it is of the form C,Xi'»+(7iXi™-iX2+... + (7^Z2-,
the C"s being numerical, hence equating coefficients of x{^ in the equation (I) we find
C^a{^ = iLX where X does not depend on b.
Consequently /n does not depend on b and therefore on making 6 = 0 in (I) we find
CoXi™ = ;x<^, for X^ is now zero.
Hence /x is constant and (/> is a numerical multiple of Xj"*, i.e. of
Thus a single linear form has no invariants and the only covariants are powers of the form itself.
We shall now assume that a covariant of any number, less than n, of linear forms which is linear in the coefficients of each form can be expressed in terms of invariants of the type (a/3) and covariants of the type ax.
Let <^ be a covariant of the same nature of n linear forms ««» ^x) yxj 5*. ."J
G. & Y. 3
34 THE ALGEBRA OF INVARIANTS [CH. II
let 01 be the result of putting ^=a in (^, (^2 ^^e result of putting y = a in (f)^, and 80 on, so that 0„_i is the covariant of the single form Og obtained by making
a = ^ = y=...
in (j).
Now since 0 is linear in /3 we have
and in like manner
'^'^V'hi^"'^)''''
etc.
Consider (a, g- + o^ ^l) <^ = 0i
as a differential equation for (f>, it being given that 0i does not contain 3. As a particular solution we have
where *S' is the degree of (^j in a, i.e. S= 2 in our case, therefore
where O^ ) =^i^ +^29^,
and further yj/^ is linear in ^.
Accordingly ^^1 = ^1^1+ P2&2 » where F^, P.^ do not contain ^ and aiPj+a^P^^O,
• ■ „ ^ ^1'
02 «i
where xi does not contain /3 and is integral in a, y, 8, etc. Hence V'i = A^i+A^2 = («)3)xi
and <^ = | (^ 9^)01 + («^);tl•
But 0 is a covariant, and ( ^ ^ j 0i is a covariant ; therefore {ai3) xi is a
covariant ; again, (a^) is an invariant, therefore xi is ^ covariant. Moreover since xi(°^) i^ linear in a, ^, y, S, ..., xi is linear in y, S, ... ; therefore by hypothesis it can be expressed in tenns of (yS), y^, 8^, etc.
Thus (f) = -(^^\(f)i together with an expression depending only on (a^), Qxy and factors of these types.
37] THE FUNDAMENTAL THEOREM 35
Then in like manner
therefore * = 2^(^a-a) (^Da) ^^
together with terms of the required form. ,
Proceeding in this way we finally have
*4(4)(ra^)('a^)-*.
together with terms of the required form.
But ^„ being a covariant of a^ is a numerical multiple of
a."
and therefore (^ 8^) (v ^) (4) * " ' '^"
is a numerical multiple of a^^xyx ••••
Thus we have expressed the covariant 0 completely in terms of the two types of factors (a/3) and a^, that is, if the theorem is true for less than n forms it is true for n forms ; but it has been proved true for one form, hence it is true universally.
Q. E. D.
3—2
CHAPTER III.
POLARS AND TRANSVECTANTS.
38. Two sets of variables
Xi, ^2 > • • • ^n )
yi,y^., ••• Vn,
there being the same number of variables in each set, are said to be cogredient, if, when one set is transformed by any linear trans- formation, the other set is transformed by the same transformation ; thus if a?i, a^j, ... x^ become Xj, X^, ... X^ where
•^1 ^^ ^1,1-Ai + 'i,2 A2 T ... "I n, n-An X2 = ^2, 1 -^ 1 > ^2, 2 -^ 2 "I" • • • I ^2, w ^ n
.(I),
then 2/1, 2/2,
•^n — 'n, 1 A 1 + In, 2 A 2 4" ... "T l'n,n-^n
yn will become Fj, Fg, ... Fn where
2/1 = ^1,1 -t 1 + ^1,2 ^2 + ••• H" n.n-'^n 2/2 == ^2,1 ^1 + %2 -''^2 H" • • • + ^2,n ^ n
.(11).
Two sets of variables
Xi, X2, ... ^n 2/1.2/2, ... 2/n
are said to be contragredient if, whenever the first set is transformed by the equations (I), then the second set of transformed variables Fi, Fg, ... F„ is given by the equations
1^1 = ^1.12/1 + ^2,1^2+ ••• +ln,iyn -» 2 = ^1,22/1 + ^2,22/2 + • . . + l"n,2yn
l^n= ^i.n2/i + ^2,ny2 + • • • + In.n^n,
.(III).
38-39] POLARS AND TRANSVECTANTS 37
It is easy to see that if the set of variables x^, x^, ... x^ is contragredient to y-^, y^, ... yn\ then 3/1, 3/2, •.. 2/n is contragredient to iTi, x^, ... Xn. For example the symbols a and a; in a symbolical product are contragredient.
39. From these definitions we deduce the following theorem :
If x^, X.2, ... Xn; y-i, 2/2 > ••• Vn (^^'0 two contragredient sets of variables, then
x^y-i + x^y^ + . . . + ^n1/n
is unaltered by any linear transformation.
Conversely, if
^i2/i "t" ^2^/2 + . . . + ^n^n
is unaltered by any linear transformation, the two sets of variables x-i, X2, ... Xn', 3/1, 2/2, ••• yn cc'^s contragredieut.
Let the two sets of variables be contragredient, then if (I) and (III) be the equations of transformation,
a-'i 3/1 +^23/2 "r ••• -^ ^nyn — ■Vi {U,\ -3^1 + ^1,2 ^2+ ••• +^l,7l-X',i)
+ 3/2 (^2,1 ^\ + ^2,2 -3^2 + • • • + 4,n ^n) +
I Vn ('»i,i -^1 + 'n,2 -^2 '^" • •• "^ ''n,n ■^n) = ^1 (^1,1 3/1 + ^2,1 3/2 + ... + ln,xyn) + ^2 {li,2 Vi + 4,2 3/2 + • • • + ^n,2 Vn) +
-\- 2Ln ('1, 71 3/1 + ^2,w 3/2 + ••• + ln,nyn)
= Xi7i+X2F2+... + Z„F„. Q.ED.
Conversely let
«i3/i+ ^23/2+ ... +Xnyn
be invariantive, then if «i, x^, ... Xn are transformed by equations (I)
Xi Y^ + X2 Y.2+ ... + Xn Yn = x^yi + ^23/2 + . . . + x^yn
= 3/1 (4,1 ^1 + 4,2 X^+ ... + 4,n -^n) + 3/2 (4,1 X^ + 4,2 X2 4- . . . + 4,n Xn)
+
+ 2^»(^n,l -X"! + ^^^ 2 -X'2 + . . . +^„_„X„).
r^w
6y9(>3
38 THE ALGEBRA OF INVARIANTS [CH. Ill
But this is an identity true for all values of Xi, X^, ... Xn\ hence the coefficients of Xj, X^, ... Xn on the two sides of the identity are equal ; i.e.
Yl = ^1,1 2/1 + ^2,1 3/2 + • • • + ^n,l Vn Y2 = 11,2^1 + 12,2^2+ •-■+ ln,2yn
^n — ^1, ra Vl + %m 2/2 + • • • + ^re,n Vn-
This set of equations is the same as (III), and hence the two sets of variables are contragredient. Q. E. D.
It may happen that a set of variables x^, x^, ... Xn is subject to a restricted group of linear transformations, and that
Xiyi + ^2y2+ ••• +Xnyn
is an invariant for all transformations which are allowed. The above proof still holds that y^, y^, ... 2/„ is contragredient to x^, x^, ... Xn. But yi,y-i, •-. yn is subject to a restricted group of transformations.
Ex. (i). If the binary quantic become after any linear transformation
(-^OJ -^15 -^2* ••• -^ni-^i, -A2)",
then (tto, «!, ... ctnl^i, •^2)"=(^o> ^i> ••• ^n$J^n -^'2)"
Hence
■^OJ ( I j -^n (2]'^^' *■■' '^'"
•*'! > "^l •''2> •••» •*'2 »
are two contragredient sets of variables, subject to a restricted group of linear transformations. The linear equations connecting the original with the trans- formed coeflBcients of a binary form may be deduced from this fact (cf. § 16)*.
* By means of the group here indicated binary invariants and covariants are brought under Lie's general theory of invariants of continuous groups. If we pass from the variables fj, ^^, ... tm to ^^^ variables fj', fj'* •■• fm' hy means of the transformations,
fp'=/p(fl, fs' ■■• imJ <*l' «2» ■•• «r)» and these transformations form a group when the parameters a vary, then a function
39-40] POLARS AND TRANSVECTANTS 39
Ex. (ii). Two sets of variables both contragredient to a third set are cogredient with one another.
Ex. (iii). If the determinant of transformation of a set of variables be /x, the determinant of transformation of the contragredient set is - .
40. If F(ao, tti, a.2, ... a;^, x^) be any covariant of a binary quantic (uq, a^, a^, ... a^^x^, x^''^ and if y-i, y^ be a "pair of variables cogredient with x-^, x^, then
y^dx^y^dx^
is unchanged by any linear transformation of the variables, except that it is multiplied by the same power of the determinant of transformation as that by which F is multiplied after trans- formation.
For if
F{Aa, A^, A„, ... Xi, X^) = fiF{ao, a^, a^, ... x^, x.^
where A^, A^, A^, ..., X^, X^ are the transformed coefficients and variables, and if z■^, z^ be any pair of variables cogredient with x-^, x^ which become Z^, Zo after transformation, then
^(^0, ^1, ••• Z„ Z,) = fMF(ao, ai, ... z„ z,), fi being a power of the determinant of transformation.
But Xi + Xyi, X2 + \y2, \ being any constant, are a pair or variables cogredient with x\, x^, hence
F{Ao,Ai,...,X^ + \Y,,X. + XV.^ = fiF(aQ, ai,...,Xi + Xy^, x^ + Xy^).
Now F being a rational integral algebraic function of all its variables, we may expand each side of the above equation in powers of X, ; since A. is arbitrary, the coefficients of the different
is said to be an invariant of the group if
^(fi'. fa'. •■•fm') = -P'(fl.f2.•••U• In our case the variables are (n + 3) in number, viz. a„, aj, ... a„, Xj, acj, and the transforming equations are all linear. The parameters are the coefficients in the original transformation of .Tj, x^, and if they be so chosen that Iii7.2~^2'?i = l- ^^^^ an invariant or covariant of the binary form is an invariant of the above group in (n+3) variables. The reader will find it interesting to write down the actual transformations for the a's and thence to verify that they form a group. Cf. Lie, Vorlenurifien ilber contiimierliche Gruppen, p. 718 etc.
40 THE ALGEBRA OF INVARIANTS [CH. Ill
powers of \ must be the same on the two sides of the equation. Hence, using Taylor's theorem :
F{Ao, ^1, ..., Zi, X2) = fMF(ao, tti, ..., Xi,x^)
( ^i^Y + ■^^ov^ ) -^(-^O) -4i, ..., Xi, X2)
from which we see that f 3/1 x-;- + 3/2 ^— ) -^ is invariantive, if F is
\ OOl^i OtX/2 1
a covariant.
41. The definition of covariants may now be extended thus : Any function of the coefficients of a single quantic, or of a simultaneous system of quantics, and of sets of variables, — all sets being cogredient with the variables of the quantics, — which is such that, when any linear transformation is made and the original coefficients and variables replaced by the transformed coefficients and variables, it is unaltered except for a factor de- pending on the coefficients of transformation.
Let a^i, 5^2 ; 2/1, 2/2 be two cogredient sets of variables, then
{x^y^-x,,y:) = {xy)
is unaltered, except for a factor — which is the modulus itself — by any transformation.
Hence, if it be necessary to consider covariants with sets of cogredient variables, we may replace all the sets of variables but one by the coefficients of linear forms added to the system. For we may regard {xy) as a linear form — since it becomes (ZF) by transformation — and hence replace the variables y^, —y^ by its coefficients.
Hence covariants having more than one set of variables may be represented symbolically as a sum of products of symbolical factors of the types (a/S), ««, Uy, {xy), —
42. Convolution*. The word convolution is used as a name for the process of obtaining from a given symbolical product (representing a covariant) another symbolical product
* German Faltung.
40-43] POLARS AND TRANSVECTANTS 41
(representing another covariant) by removing two of its factors of the form cLx^x and replacing them by a single factor of the form (ayS). Any symbolical product P' obtained from the product P by means of this process either once or several times repeated, is said to be obtained from P by convolution. Thus the covariant {ahf{ac)aJ)^Cx of the quartic is obtained by convolution from the covariant {ahfa^b^Cx*; the factor a^Cx being replaced by {ac). It may also be obtained by convolution from {ah){ac)axhxCx', also from {ah)axhxCx*, or from axhxCx-
This process, it should be noticed, is purely a symbolical process, and has no analogue in the non-symbolical treatment of modern algebra.
43. Polars. The operator
has already been introduced. In § 15 we defined the form
^'(^a^)> (^-)
to be the rth polar of P ; P being a binary form of order n.
Again in § 40 of this chapter it was proved that if P is a covariant and the variables 3/1, y^ are cogredient with x-^, x^, then the form (IV) is also a covariant.
For the purpose of calculating polars, we may use a theorem identical with that of Leibnitz for ordinary differentiation.
Thus if the ?'th polar of a^h^
be required, it may be obtained by operating on this expression with
(m + »)!
where Pj = y^ ^ — + 2/2 ^ , but operates only on ax'* ; and Pg is
the same expression but operates only on bx^.
Ex. (i). Find the .second polar of (ab)- ajbj. Ex. (ii). Find the rth polar of a^'^h^^cjj'.
42
THE ALGEBRA OF INVARIANTS
[CH. Ill
44. In view of the fact that covariants will generally be given in terms of symbolical letters which refer to the original quantic or quantics, and that in this case the factors a^ are not all the same, we shall consider the general case in which these factors are all different. Results, proved for this case, may be obtained for any other by simply equating two or more letters.
Consider now the form
where P is a product of symbolical factors not containing x-^, x^. The first polar is
" fl V ' ' X
|
+ |
||
|
Is n |
•••««-!, "nj |
the term in the bracket being simply an abbreviation for
We notice in this expression that :
(i) The difference between two terms
L «r
-F'^.oi,^
y " -
]
L «r a« J where X is an expression obtained from F by convolution.
(ii) The difference between the whole polar and one of its terms
F''.ai^-^
i^*".a.
POLARS AND TRANSVECTANTS
43
44-45]
where X is a sum of terms each obtained from F by convolution, each such term being possibly multiplied by a constant.
Similarly the rth polar of F^^ is
fli otg . • . Or
F.
a, cCi
(V),
where the summation extends to all possible sets of r factors taken from a.-^ a-i ... a« . To see the truth of this suppose it true for the rth polar, then the r + 1th polar may be obtained from
the 7'th by operating with — • (y^)- ^o the left this gives
n — rX' dxj
■p^n-r-i p^r+i^ Consider one term on the right, that expressed above ; by polarizing we obtain
«! tta ... <Xr di '
r\{n-r -1)1 ^^
F^.
«! Oa
Clr °li
Each term of (V) gives rise to a similar expression. Now any particular term of the r + lth polar, arises r + 1 times, once from each of r + 1 terms of the rth polar, thus
" tti a^ ...Or Or+i " J/T n y y » »
arises from each of those terms of the rth polar, obtained by omitting one of the factors in the numerator and the corresponding factor of the denominator of this expression. Hence
ai fla • • • O^r+i ^ , y y _y
aj^Og^ ... a,.+i^
Now it has been seen that this is the correct form for the first polar ; hence it is the correct form for the second, and so on ; the formula is therefore true in general.
45. The number of different terms
^ «! "2 • • • Or ^
FJ^ ' " -'
pn-r-. p ,+, ^ (r+l)!(»-r-l)! ^
" n !
F^
18
IS
The coefficient of each term as it appears in the polar
F^^'-^F/ 1
0'
44 THE ALGEBRA OF INVARIANTS [CH. Ill
hence the sum of the coefficients of all the terms of a polar is unity.
This remains true if some of the letters a become equal, for no term of the polar can vanish.
46. Two terms of the rth polar are said to be adjacent, when thev differ only in that one has a factor of the form a^ aj, while in the other this factor is replaced by a^ a* •
We shall now prove that :
(i) The difference between any ttvo terms of the rth polar of F^ is equal to {xy) X, where X is a sum of terms each of which is a term of the r — 1th polar of an expression obtained from F^^ by convolution.
(ii) The difference between the rth polar of F^ and any one of its terms is equal to (xy)X, where X is a sum of terms each of which is a term (multiplied by some constant) of the r — 1th polar of an expression obtained from, F^ by convolution.
The difference between any two adjacent terms
%\ F, - ccha^^ F, = (xy) (a^afc) F^ = (xy) X
" ' , r i?!," ~ .
where X is a term of the ?•— 1th polar of (a^ajk) ^ ~ , *'-6- a
term of the r— 1th polar of an expression obtained from F^ by convolution.
Now any term of the rth polar of F^'' may be obtained from any other by means of a finite number of interchanges of letters such as ail, aj^. Hence between any two terms Ti, T^ of the rth polar a series of terms Tj^,, Tj^j, ... T^^i may be placed such that each term of the series
-^1) -^1, i> -^1,2) ••• -'i, I) ■'a
is adjacent to that on either side of it. Hence the difference between two terms
T,-T,= {T,-T,,,) + (T,,,-T,,.} + (T,,,-T,,,)+ ... +(T,,i-T,)
= (xy)X
where X is a sum of terms each of which jis a term of the r — 1th polar of an expression obtained fiom F^^ by convolution.
45-47]
POLARS AND TRANSVECTANTS
45
Again, the difference between the complete polar and any single term T
Hr)
= 2
i = l
T,
-T
Ti representing the general term in the polar
'T,— f
Hr) S
and hence the theorem (ii) follows at once by (i).
It should be observed that if two or more factors are now made identically equal, the above proof is not affected ; we may for the sake of argument suppose them all different, and make them equal in the final result. The only effect of the equality of factors is that some of the terms obtained by convolution from Fx^ will vanish. The propositions are true, then, as stated, for any symbolical product ; and consequently for any covariant form.
47. Let T be any term of the rth polar of F^^, then by proposition (ii) of the last paragraph
where ^^_i is a term of the 5 — 1th polar of a form obtained from Fx^ by convolution and \y_i is nurnerical. Let the r— 1th polar, of which (jjr-i is a term, be -v|r,._i. Then applying (ii) again we have
(f)r-i — ^{rr-i = i^y) 2 Xr-2 4>r-2
where ^r-2 is a term of the r — 2th polar of a form obtained by convolution from F^^.
Proceeding thus we see that
where ■yjr^. is the Jcth. polar of a form obtained from F^^ by con- volution ; and A^ is numerical.
46 THE ALGEBRA OF INVARIANTS [CH. Ill
The terms of this series, the existence of which has just been demonstrated, will be accurately determined later. The series is known as Gordan's series.
48. Transvectants*. If aa;"*, 63:'' be any two binary quantics,
the form
{aby a/"-*- 6/-*-
is called their rth transvectant, or their transvectant of index r. The symbol
(/ <f>r
is used to denote the rth transvectant of two forms/, <f).
Thus
(a/'^, 6/)'- = {aby a^-^ b^\
The definition of a transvectant just given is symbolical : the process of forming a transvectant is however not a purely symbolical one, — like that of convolution. In order that we may be able to obtain transvectants of any two forms, we make use of the diffe- rential operator, introduced in Chapter I.:
n = -^^ ^.
Thus
fn, t M !
" {m — ry. {n — ry. ' ^
Hence if f{x\ (f> {x) be any two forms of orders m, n re- spectively, then the rth transvectant of / and (f> may be obtained
by operating with ^ 1 -^ — r- ' Cl^ on f{x).<f>{y) and after
operation replacing y by x.
Thus
{m — ry. (w — r)!
{f{x), <!> {X)y = ^^^^- . ^^^- [nrf{x) . (/> (2/)],., .
Ex. For the cubic the first and second transvectants of the Hessian with the cubic itself are
{{ahf a^b^, c^sy=^ {ahf {ac) Kc^^-^\ {abf {he) a^c,\
{{ahf ax hx , c^^f = {ahf {ac) {he) Cx ;
as may be seen by using the differential operator.
* German Vberschiebung.
47-49] POLARS AND TRANSVECTANTS 47
It is useful in calculating transvectants to notice that, just as in the case of polars, the sum of the coefficients of the various terms of a transvectant is unity.
49. For the purpose of calculating transvectants the following method is extremely useful.
Consider the rth transvectant of two forms aj^, h^\ it is {ahy a^"*-*" h^-^.
It may be obtained by the following rule :
Polarize a^^ r times with respect to y, we obtain a^^'"^ aj \ then replace y-^ by h^, y^ by — 6j and multiply by ho^~^, the result is {ahy »«:"'"'■ &»''"'■ which is the rth transvectant of a^ and h^.
We proceed to illustrate the method :
(i) Consider the second transvectant of the Hessian of a quartic with the quartic itself,
The second polar of (ahy a^b^^ is
^ (aby ay^bj^ + | (aby ajy^ayby + ^ {aby aj^by^ Hence the transvectant required = 1 {aby (acy b^'cj" + 1 (aby (ac) (be) a^b^c^^ + J {aby {bey a^d = \ {aby {acy bj'c^' + | {aby {ac) {be) a^b^c^^ since a and b are equivalent symbols.
(ii) The third transvectant of these two forms is {aby{acy{bc)b^c^.
(iii) To obtain the second transvectant of the Hessian of the quartic with itself: i.e.
{{abya,'b,\{edye^'d^J.
Let us write {cdy c^di = hx*, where the symbolical letter h refers to the coefficients of the Hessian considered as a separate binary form. Then as in (i)
{{abya^'b^Mi^J
= J {aby {ahy bJ'K^ + 1 {aby {ah) {bh) aJ)^K\ To obtain the first term we polarize h^* twice, replace y^ by — a^
48 THE ALGEBRA OF INVARIANTS [CH. Ill
and 2/2 by + aj and then multiply the result by ^ (aby bx'- Thus the second polar of h^*
= Hcdy Cy'dJ' + 1 (cdy CydyC^d^ + J (cdf C^Hy\
Hence
\{aby{aKfbx'K'
= jg {ahf {acf (cdy b^H^ + \ (aby (ac) (ad) (cdy c^dj)^
^^^(aby(cdy(adybx'c^
To obtain the second term we polarize h^* once with respect to y and once with respect to z, and then replace y hy a and z by b.
The polar is
^ (cdy CyCzdi + \ (cdy CydzC^d^ + \ (cdy CgdyCxd^ + ^ (cdy c^dyd^.
Hence
\ (aby (ah) (bh) a^b^hj" = ^ (aby (cdy (ac) (be) ajy^d^^
+ I (aby (cdy (ac) (bd) aj^c^dx + f (aby (cdy (ad) (be) a^bxC^d^
+ ^ (aby (cdy (ad) (bd) a^b^d.
Hence remembering that all four letters are equivalent, we obtain
((abya^^bx\(cdycHry
= \ (aby (acy (cdy b^^d^ + 1 (aby (cdy (ac) (ad) c^dM + f (aby (cdy (ac) (bd) aJ)xCxdx. (iv) Calculate
(v) Transvectants of the following form are of frequent occurrence :
= 2 Uy \m/ ^^^a ^j^ n-Xj^m-^ p-r
The result may be obtained at once by polarization, (vi) From this may be deduced the value of
Let Cj5Pc?,9=Aa,P + «.
49-50]
Then from (v),
POLARS AND TRANSVECTANTS
49
T= 2
<T-Vr=r
CKI)
{aKf {hhy a/- " h^"^-^ h^P+^-''.
But polarizing Ax^^* f times with respect to y and r times with respect to z we have
^ ^ X!.!(^-X-.)! ,!z.!(g-^-.r)! ,
cr\ t! (^ + g'-o- — r)! Hence replacing the y's by a's and the /s by 6's
P!
r =
2 ?•! (n-X-^)! (m-i/-tzr)! (;?-X-i/)! (y-/x-ar)!
{'m + n — r)\' {p + q — r)\ (vii) If /=ai a2^...a^ , 0=^, ^2,... /3„ ,
(/, <^r=
1 sf (ai/3i) (02/32) -K^r) ^ ,1
where the 2 extends to all possible arrangements of the letters aj, 02, ...a„; ^i,)3„..A. §44.
50. Two important theorems relating to the difference between terms of a transvectant, must now be proved. They are exactly analogous to those already obtained for polars in § 46.
(i) The difference between any two terms of a transvectant is equal to a sum of terms each of which is a term of a lower trans- vectant of forms obtained by convolution from the original forms.
(ii) The difference between the whole transvectant and any one of its terms is equal to a sum of terms each of which is a term of a lower transvectant of forms obtained by convolution from the original forms.
Here, as in the case of polars, we introduce the idea of adjacent terms. Two terms of a transvectant are said to be adjacent when they differ merely in the arrangement of the letters in a pair of symbolical factors. Two terms can be adjacent in any one of the following ways :
(i) P (ai^j) (a^^k) and P (a,/8,) (a^^j), (ii) P (ai^j) tth^ and P (ah^j) \, (iii) P (ai^j) ^[ and P (a^/S,) y8-^,
G. & Y. 4
50 THE ALGEBRA OF INVARIANTS [CH. Ill
where the letters «!, otg, ... belong to the first of the two forms in the transvectant, while ^i, ^2,--- belong to the second form.
The difference between two adjacent terms is in the three cases seen to be
(i) P(«,ar,)(/3,-/3,),
(ii) P{aia„)^j^,
(iii) P(^,^j)ai^.
To fix ideas we shall suppose that the transvectant we are considering is the rth transvectant of
and (f) = B.^^^^^^...^n^,
where A and B are products of symbolical factors of the type (yB), and all the factors of the type y^ are different.
Then
(/.<^r= ^
C)
where the S extends to all possible arrangements of the letters tti, Oa-.-o^; /3i,^2-..^n,— § 49 (vii).
The truth of this statement may also be seen by operating with Xl*" onf{x) 0 (y) ; and remembering that if we write
ai = a2= ...=a^ = a, /3i =^2= .•• =yS„ = /8,
then (/ (f>y = (a^y a,"^-^/3^''-^.
The difference between two adjacent terms of the above trans- vectant is a term in which at least one factor of the type (ayS) is replaced by a factor of the type (oca'), or else of the type {^fi'). There are then not more than (^ — 1) factors of the type (ay9).
Hence the difference between two adjacent terms is a term of a transvectant of index less than r, of forms obtained by convolu- tion from / and <j).
Thus, for example,
(abf {acf h^ c^, (aby (ac) (be) aja^c^
are adjacent terms of the transvectant
{{abya^^bi,c^y, and their difference
{aby(ac)b^c^^
50-51] POLARS AND TRANSVECTANTS 51
is a term of
{{abfaj)^, Co;*).
Now between any two terms Ti, T^ we may place a series of terms
-'i, 1> -'i, 2» ••• -^ 1, i
such that any term in the series
1> -*^1, 1> -^ 1, 2> ••• -^ 1, 1> -^2
is adjacent to that on either side of it.
For we may obtain T^ from Ty by a finite number of inter- changes of pairs of letters, a pair being composed either of two as or else of two yS's ; since each letter occurs only once — in our argument, — the terms which differ by the interchange of a single pair are adjacent. Hence the difference between any two terms
i.e. a sum of terms each of which is a term of a transvectant of lower index of forms obtained from the original forms by convolu- tion. This is the first theorem.
Again if T be any term of the transvectant, then
(/, (jiy-T= — !— - tr - T
r) \r.
2 {T - T)
[since the number of terms T' is, r\{ j ( j J
and this is equal to a linear function of terms of transvectants of lower index of forms obtained by convolution from y and <j>.
51, This theorem may be extended by applying it again to each of the terms of transvectants of lower index on the right- hand side. This may be done repeatedly. Each time the process is applied the terms of transvectants on the right are replaced by the transvectants themselves, and linear functions of terms of transvectants of lower index. After not more than r applications
4—2
52 THE ALGEBRA OF INVARIANTS [CH. Ill
of the process, we have on the right a linear function of trans- vectants of forms obtained from f, <f> by convolution, whose index is less than r, and of terms of transvectants of zero index.
But a trans vectant of zero index of two forms is simply the product of the forms ; a term of such a transvectant is merely the same product, and therefore the transvectant itself
We obtain then the following important theorem : The differ- ence between any transvectant and one of its terms is a linear function of transvectants of lower index of forms obtained from the original forms by convolution.
Ex. (i). {abY(bcya^W
= {iabf a,^h,\ c,*)2 - % {ahY (be) a,cj^ - J {ahf (ac) b^c^^ = ((a6)2 a^^bj', c^^Y - (Sflbf a^b^, c^*) + § {abf . c^\
Ex. (ii). Prove that
{{ab)a,^bj^,{cd)cj^d,^f
= ^iab) (cd) {{boy {adf+9 (bcf {adf {ac) (bd)
+9 (6c) (ad) {acf {bd)^+{acf {bd)^}
= J {(&«)* («^* - («c)* {bdf} - \ {abY {cdf {{bcf {adf - {acf {bdf}
= i {Q>cf (ad)* - {acY {bdf) - \ {abf {cdf {{be) {ad) + {ac) {bd)}
= J {{be)* {ad)* - {ac)* {bd)*} - 1 ((a6)3 a,b, , {cdf c^d^f.
Ex. (iii). If /be a cubic and H'\\& Hessian, prove that {H,ff=0.
Ex. (iv). If/ be a quartic and ZTits Hessian, prove that {ff,f)^ = 0.
Ex. (v). If bx* be the Hessian of a^*, and {ac)* = 0 = {ad)*, then
{{ab)a^^bxMcd)ex^dx^f=0.
Ex. (vi). The second transvectant of a quantic / of order n and its Hessian II is
where i={f, /)*.
For
V 2 )
= (a*)2(ac)2 a,'»-46.»-2o.»-2- gj^^ t/ on using (III) § 22.
Finally the result is obtained by the help of the last identity of § 22.
CHAPTER IV.
GORDAN'S SERIES.
52. Gordan's series. In § 47 it was proved that any term T of the 7wth polar of a symbolical product F can be expressed in the form X{xyyFi; where Fi is the (m — i)th polar of a sum of symbolical products — possibly multiplied by positive or negative constants — each of which is obtained from F by convolution. The product a^hy^ is a term of the with polar of ax"})^, hence
a,«6,- = 2(a;2/yi^<V-* (I);
where the suffix indicates that F'^^^^-i is the (m — i)th polar of F^K
This is an identity; it must therefore be true when y = x*\ hence
for when y — x, {xy) = 0. Now if in a function of x polarized with respect to y, y is replaced by x, the result is the same as the original expression. Thus the rzth polar of a,^'^ is a^ay^, which becomes a^'^'^ when y is replaced by x. Hence
a^H,-^ = \F<\m\^^=F' (II).
Thus the first term of the series (I) is obtained, the other terms may be obtained in a similar manner. Operate on (I) with IP'. Consider first the effect of this operation on the term
{xy)'F^\,.-i. * This of course means — = — .
54 THE ALGEBRA OF INVARIANTS [CH. IV
By comparing the degrees in x on the two sides of (I) it may be seen that the degree of F^^^^-i in ^ is n — i. Hence we may write
and F^^ym-i = a^^-'a/'-\
Now
= i(m+n-i + l) {xyf-^(i^-^(iy^-\ a result obtained by ordinary differentiation as in §§ 26, 27. Hence by repeated operation of VL we obtain
\ y' ^ y {^-J)\ {m + n-i-j Jriyr ^' ^ v '
when i >j; but when i <j
nJ {xyfaj'-^oiy'^-^ = 0.
As regards the right-hand side of the equation (I) we know, § 27, Ex. (i), that
nJaJ'by'^ = ; ~, . 7 ^, {ahya^^-iby^-i.
(n-jy. (m-jy. ^
The result of this operation is, then,
in-jy.'im-jyr^^'''' ^'
= 2 y. — TT^, -. — -. — =-r-, {xyf •? J^'*V-<.
i=i (t -;)!(w + w - 1 -J + 1)! ^ ^^ ^
In this put y = x, and we obtain
(w-^)! (m-;)!^ * * (m + w-2j + l)!
and hence
F(3) = ^J^^3/ {abya^^'-^hx'^
in
52-53] gordan's series 56
Substituting this value of ^'■^"' in (I) we have Gordan's series : —
^n\ /m i=o /m + n — i
/m + n — i + 1\
53. This series may be put into other, rather more general, forms.
Multiply by {aby, write « + ^ for n, and m + r for m, then
«-.--.(M^)«<..-.«;.
Operate with ix-^], and we obtain
^n — r\ fra — r — k i
fn — r\ fi
Operate with [y ^] on (IV), then
m—r
'y
^n — 1 — k\ fm — r
= 2
+„-2;-.>i\("^>'('"" *«-);:-— - (^i)-
In (V) replace y^ by Cg, 3/2 by — Ci, and multiply by c^P"^'^^'^^, we obtain — see § 49 —
'n — r\ fm — r— k
7)C
= X{-iy " " ' \ — V-^ ((aa;^ bary^'-, c^^)"»-*-^-*...(vii).
56 THE ALGEBRA OF INVARIANTS [CH. IV
The equation (VI) gives a similar result when the weight of the covariants under consideration is greater than m the order of fea,*": viz. —
(aft)*- (6c)'™-^ (ac)*aa;''-^-*Ca;2'-"'+'-*
/n — r — k\ /m — r\
= S ( - 1)^' L + L2r-/+l( ^^^»''^' ^*">'^' c.^)-i-+* . . . (VIII). \ i J
54. Now if in (VII) a and c, n and p, r and m — r — k are interchanged, the left-hand side of the relation is unaltered, except for a factor ( — 1)"^~*. Hence
fn — r\ .Vfi — r — k\
fm + n — zr -i+\\
'p + r-'tk — m\ fr\
= 2 ^ ^ ^ . :, ^-^-frv ((^*"'. 0^^)^+"^'--*, a*")''"*-
'^ + 2r-l-2« — w — t + l\
On the supposition that
f=h^^, <i> = aar, '^^cJP, a, = 0, a^ = m-r-k, as = r, this relation is the same as
% A ! l\ll^((f A,)<H+i y}rY: + <h-i ,
= ( - 1)"' S ^^ \ .\ (( /, ^|r)-.+», <f,)».+«3-* . . . (IX).
^ ^ /m+p - 202-1 + ly^-^' ^^ ' ^^ ^ ^
Again, if in (VIII) we interchange a and c, n and p, r and m — r respectively, the left-hand side of the relation is unaltered except for a factor (— 1)"»"^.
Hence we obtain a relation of the same form as (IX), but in which
ai = k, 0Lz = m — r, ct^ = r.
53-55] gordan's series 57
Thus the identity (IX) is true in two cases. In the first ai = 0 : as cannot exceed either m or n, for we use in (VII) the factor {aby : 0^+ cis — m — r and therefore cannot exceed m : and finally a^ cannot exceed p.
With these restrictions Oa, a.^ may take any positive integral values.
In the second case
tta + as = ini-
ttj + «! cannot exceed n, and a^ + aj cannot exceed p ; subject to these restrictions a^, a^, a^ may take any positive integral values.
The series (IX) is one of great importance for calculating transvectants. It is usually quoted in the form
/ f <f> f \ { m n p ) ,
\ «! 02 Ots /
«•> + Ofs > W, tts + fli > n, «! + tta > p,
where
and either
(i) fli = 0
or
(ii) aa + Qs = wi.
55. In order to illustrate the use of the series, we shall now calculate some transvectants which are covariants of the sexticf*. Let us write
H = {f,f)\ i = (f,fy, A=(f,fy, t = {f,H). Then to calculate the transvectant (H, fy we use the series
/ / / / \ (666), \ 0 3 2 /
(7) (7)
* other examples will be found in Chapter V.
68 THE ALGEBRA OF INVARIANTS [CH. IV
that is
{H,fr+^{i,f)={i,f),
or {H,fr = \{i,f).
The object usually in view when calculating transvectants, is to express them in terms of other transvectants of lower index.
There is no difficulty in choosing the series which will give the desired result, in general we select it so that the transvectant to be calculated appears as the first term on one side of the identity, while all terms on the other side are of lower index.
The transvectant (H,/)* is given by / / /
/ / / / \ (666) \ 0 4 2 /
((/ /^^ fy-' = ^4P4 ((/ n*^'' /)'-'>
hence
CT) {'?)
iH,fy + ^-^{i,fy+lA.f=(ify + lA.f,
or {H,fy = ^(ify + ^A.f.
To calculate (H, fy we cannot write ai = 0, for then the condition a^ + aai^m would be violated.
We must then make ttg + ctj = m = 6 and use the series / / /
// / / \ (666) \ 1 4 2/
hence
i ) K i I
(ff,/)'=-y*'(i,/r.
* This form vanishes identically — see Ex. (ii).
55-57] gordan's series 59
To find (t, fY we use
// H f. 16 8 6) \ 0 2 1 /
whence using the result, § 51, Ex. (vi) we obtain
Ex. (i). Prove that for the sextic
(zr,/)«=^(t,/)*.
Ex. (ii). By means of the series
/ / /
6 6 6
13 3 prove that (i, f)^ = 0.
Ex. (iii). Prove that
iif,^r',ff=-^QU^r,n
{H, ^•)4=2^^•,^•)2 + ^^.^•.
56. The series we have been illustrating does not give all the relations between the covariants of/, ^, -^ which are of weight
«! + «2 + "s-
For example, we cannot by means of this series reduce the form ((/, fy, fY which is reducible whatever be the order of/, § 51, Ex. (vi).
57. The series (III) may be inverted with the following result
n («)
(a,™, 6=.»)V = S(- l)*^)£;^(aJ)'a,"'-'6„»-«(^2/)' (X).
i I m "T '^^
\ i J (Gordan, Invarianten-theorie, § 7, pp. 89, 90.)
60 THE ALGEBRA OF INVARIANTS [CH. IV
To prove this, we first establish the existence of an expansion of the form indicated ; and secondly we find the coefficients. To prove the possibility of such an expansion, we observe that
= l\ia^^-%''-^ [aj)y - {ah) {xy)Y = S/Uj {aby {xyy a^-i hy^'-i.
To determine the coefficients /u,, we operate on the relation just proved with ft ; the left-hand side of the relation vanishes, and hence by equating the coefficients on the right to zero we obtain
yLtii (m + w — *■ + 1) = - /ii_i (m — I + 1) {n — i-\- 1).
Replacing y by a?, we see that /io= 1, hence in general f^i = (-) 7
m + ri
Other series may be deduced from (X), by the same methods as were used in § 53.
Thus
f7n — k\ fn — k
I / \ I
(aoT, h^fyn-u = 2 {-y \^/^_2M («^)''*-*«^""'"* V"'-* (^)^
((«*"*, hx^^f, cjy
'tti — k\ (r
{
= S ^ — " ' 'V. (aby+''{bcy-' a^"^«-*&a;"-'-*Ca;^-'-+^' . . .(XI), /m + n-2kY ^ ^ ^ ^ ^
when r if-n — k.
fm + n- 2k — 7'\ hi — k\
(K"*. &*")*, cx^r = S -^ r^-r 1 , '
X (a6)»+* (6c)"-*-* (ac )*+»-" a^m+n-2*-r-tc/-r+f (XII).
58. It has already been pointed out, § 41, that the system of concomitants belonging to a binary form with two or more cogredient sets of variables, is the same as that obtained when
57-60] gordan's series 61
certain binary forms with only one set of variables are taken for simultaneous ground-forms. Gordan's series (III) shews us which the ground-forms must be ; thus for the form
we consider the system
ax^hx, {ah) ax, {xy).
Each member of this system is unaltered by linear transforma- tion, and ax^hy can be expressed in terms of the forms given.
59. Ex. (i). Any symbolical product having two cogredient sets of variables x, y, can be expressed in the form
where P is a symbolical product containing only one set of variables x.
If we put
Xy^l, .^2 = 0, yi = 0, ^2 = 1.
then Pym-i
becomes the coefficient oi x-^~'^'^^x^~^ in P.
Hence we may express any rational integral function of the coefficients of a binary form linearly in terms of the coefficients of its covariants. (Elliott, Proc. London Math. Soc. vol. xxxil. p. 213.)
Ex. (ii). Express the product ao'^i*2 of the coefficients of the cubic
(<^0> '^IJ '^2) ^X-*^!} •^2/
linearly in terms of coefficients of its covariants.
60. If a symbolical product representing a covariant of a single binary form contain a factor (aby^~^, it may be expressed in terms of products each containing a factor (ab)^.
Let the order of the binary form be n ; and let P be the covariant in question. Then since P contains the factor (a6)^^~^ it is evidently a term of a transvectant
{(abf^-'ax''+'^^bx''+'-^\ (f))",
where (f> is some other covariant.
Hence by § 51
P = ((a6)'^-iaa,"+i-^6a,"+i-=^\ (f>y
+ tC {{aby^-'ax''^'-'^ 6^"+^-^, </>)"' (XIII),
where C is a constant, and ^ denotes any function obtained from <fi by convolution.
62 THE ALGEBRA OF INVARIANTS [CH. IV
Now
for h and a are equivalent symbols. Hence all those transvectants in (VIII) in which no convolution has taken place in the first form are zero. But every form obtained by convolution from (a6)^^~^aa;"+^~^^6a;""^^~^^ has a factor {ahf^. Hence every trans- vectant in (VIII) either vanishes or is the transvectant of a form having a factor (ab)^, with another form : and hence P can be expressed as a sum of terms each of which has a factor {aby\
A convenient way of expressing this is to write
P = 0, mod {aby\
61. Owing to the large number of symbolical products which represent covariants.of given degree and order it is important to obtain methods of classifying them. For this purpose the greatest index of any determinant factor in the symbolical product is chosen. We shall use the word grade to denote this index. If the symbolical product is a covariant of a single binary form, then, by the theorem just proved, co variants of odd grade may be expressed in terms of covariants of higher even grade. On this account the German equivalent Stufe is used for half the index. We prefer to define grade as the index itself, since the classifica- tion is useful when the symbolical product is not merely a covariant of a single form.
62. Covariants of degree 3. We proceed to obtain criteria, by means of which it may be at once determined, whether or not a given covariant of degree 3 can be expressed in terms of covariants of higher grade. These were first obtained by Jordan* in 1876. They were independently discovered by Strohf; and his method of investigation is given here.
We consider first the covariants of weight w which are linear in the coefficients of each of three binary quantics
/i = a,r^ , /2 = K""' , fs = c^"'- Let us write Ui for (be) a^, Wj for (ca) b^, m., for (ab) Cx ; then
^1+^2 + ^3= 0.
* Liouville, 2 S6r. in. 1876.
t Math. Ann. Bd. xxxi. p. 444 et seq.
60-62] gordan's series 63
Hence
v^a-iui = (- Vf-i m/ (w2 + Ms)^"^
^=" (?) Multiply this expression for u-P~^u^ by
where h-^, k^ are any positive integers, and take the sum of each side of the result from i = Q to i = k2: hence
= (- 1)^^+^ i^ (^) (^ ~ ^ ~^' ~ ^) t^/--t.3^ t (XIV).
Now when
g — ky> X > g — ki — k^ — \y
therefore the last sum written down may be divided into two series, viz. \ = 0 up to g-k^-k^-l, and \ = 5r - A^i up to ^.
Let g — ki — k.2—l=k3.
Then the first of these series is (writing i for X)
In the second series, write g —i for X, then it becomes
lf^)r'-t'~^U"-v (XV).
* ( ) is the coefficient of x^ in the expansion of (1 + a;)'.
t For S (^T^) ( " \~^) is the coefficient of x*» in (l + ar)^"'*' . (l + x)-^'-l.
64 THE ALGEBRA OF INVARIANTS [CH. IV
Now if I and m be positive integers
\m) ^ ' m\(l-iy. ^ ^ \ m ) 'l-\-m-V
= (-Y , ^ ^ V ^-1
-(-r^'-'{~P_~i^) (XVI).
Hence the series (XV) becomes
|„<-)^"*-'(f)(t-~i>''""'^'
=<->'•"■!„(?) (t- /)'"'"'<"■ +'^>'
But the coefficient oiv^^~^Ui^ is here equal to ^ (9\ (9 - ^\ (- h - 1'
\,\J \i — X./ V ki — i ^ (9\ (^1 + A^s - \
Hence the series (XV) becomes
By means of the relations (XVI), the series on the left-hand side of (XIV) becomes
Hence the equation (XIV) may now be written
+(->'• I.© f'^fe "')"=""'"''=" <^™>-
62-63] gordan's series 65
This equation is true for all positive integral values of k^, k.,, k.^
for which
k^-\-ko,-\-hi = g - 1.
63. Let us suppose that w, the weight of the covariants under consideration, is not greater than the order of any of the three quantics.
Then any one of the covariants may be expressed in terms of the set
where t = 0, 1 , ... w. For since n.^ -^ w, whenever a factor (ca) appears in a covariant of this weight, it may be removed by means of the identity
(ca) hx = — {he) ax — {ab) c^.
Similarly all the covariants may be expressed linearly in tei-ms of the set
(6c)"'-' {cay a/''-* hx''^-^'-'^ c/''""', or of the set
(ca)"' - '■ {ahy a^'^ - *" h^'^- ' » c/3+« - w_
We shall suppose the members of each of these sets arranged in order according to increasing values of i. Then the forms of any one set are linearly independent. For suppose a relation to exist between the forms of the first of the above sets. Let the terms in this relation be arranged in the order indicated. The relation still remains true if we take for fy,f 2, fa special quantics instead of general ones. We shall suppose them to be merely powers of linear forms ; the result of this is that the letters a, b, c are no longer purely symbolical. Hence if
Xi {ab)""-' {bcY aa,«'+^-«' i^;"^-"' c^,-"'- '
be the first term, we may divide the identity by {bey. Every term of the quotient except the first contains {be) as a factor; but the quotient must be zero, hence the first term must vanish, when we make b = c,
ie. (aby-'-' aa;'*'+'--«' ft^w^.+ws-w'-' = 0
which is clearly untrue. Hence no linear relation can exist between the forms of one set. And therefore there are exactly w +1 linearly independent covariants, which are of the first degree in the co- efficients of each of three quantics, and of weight w (:f> 7ii, Tig or n^). G. & Y. 5
66 THE ALGEBRA OF INVARIANTS [CH. IV
64. Writing in (XVII), for Ui, u^, u^ their values in terms of a, b, c, we obtain a linear relation between the first ki + 1 members of the first set, the first A^a + 1 members of the second set, and the first ks+l members of the third set, where
ki + k2 + k3 = g — I =tu — 1.
Thus we obtain a relation between w+2 forms.
Hence if we take the first irii forms of the first set, the first niz of the second, and the first n^ of the third, where
nil + rrii + nis = w + 1,
we have a set in terms of which all other of these covariants
can be expressed. This set may be chosen so that it contains
2w no covariant of grade less than -^ .
For if w = 3m — 1, we may take
and we see that all covariants can be expressed in terms of those whose grade is 2m at least.
If w=Zm - 2, we may take
and we may express all covariants in terms of those whose grade is 2?h - 1 at least.
If w=3m. we take
mi = m + l, m.2 = 7n^ = m
and we have to include one covariant of grade 2m, the rest being of grade 2m +1 at least.
In the second case there is one relation between the 3m covariants whose grade is <|;2m— 1 ; viz. by (XVII),
(a6)2»» - 1 (6c)"* - 1 aj;»i - 2»* + 1 b^'h - 3m + 2 ^^..3 - m + 1
4- (6c)2"» - 1 (ca)"* - 1 a A " »» + ' 6a;"* ~ "'"" * '■ t'x"' - 3"' + 2
+ (oa)2»» - 1 (a6)'»~J aj:"i -3m + 2 ft^Tiu - m + 1 g^H, - 2», + 1
= 2C^ , (XVIII),
where C'2,,, denotes a covariant whose grade is <t:2»!.
In the third case there are two relations between the covariants of grade ■^2w ; these will be found from (XVII) to express that the difference between any two covariants of grade 2tn and weight Sm,
64-66] gordan's series 67
It would be sufficient for the general theory to prove the theorem : — If Uj + it, + Us = 0 then there are w + 1 linearly in- dependent products of u^, U2, u-s of order w such that each contains
an exponent of — at least : this is not easy.
65. Next let lu be greater than the order of one or more of the quantics.
If iv>ni, the sum of the indices of (ab) and (ac) cannot be greater than n^, and hence (6c) must have an index equal to w — III at least.
We define the quantity e,: to be w — Ui if w > ni, and to be zero in the contrary case. Then each of our covariants of weight w must have a symbolical factor
{bcy^ {cay^ {aby\
The remaining factor will be a symbolical product representing a CO variant of weight
W — €1 — e.) — €3 = •57,
of the quantics
^^M,-e.,-e3^ J^Mj-ea-fi^ p^^Ms-n-fj.
But the weight or is not greater than the order of any of the three quantics : hence we may apply the results of the last paragraph. That is, all the covariants of weight w may be expressed in terms of ct + 1 of them. If ei, e.^, €3 are unequal our choice of covariants in terms of which the rest are to be expressed may not be the same as before. But the student will have no difficulty in writing down the result as regards grade.
Ex. Prove that any covariant of degree three can be expressed in terms
of covariants of grade — .^ — at least.
66. We have now two different series which may be used for obtaining relations between covariants involving three symbols. These are series (IX) due to Gordan, and Stroh's series (XVII)
( - 1)*-^ % rj) (^' '^^' ~ *) (ab)^ - '■ (bey a^''' - «>+' 6/^ - '" c/^ ->...= 0,
where ki + A'a -f- A^s = w — 1
and w is not greater than any one of the numbers
r'l, ?'2, n.,. If this condition as regards lo be not satisfied we write w — til = 61, w — n., — e.i, w — ??3 = fj
5—2
68 THE ALGEBRA OF INVARIANTS [CH. IV
where it is understood that 6 = 0 if n>w. Then the reduced weight ta- is
W — €i — 62 — 63.
In this case Stroh's series is obtained from (XVII) by writing ■53- for w, Wi — €« — 63 for )h, n^ — ea-ei for n.^, ns-Ci — e^ f'^r "s, and multiplying the result by
(6c)'i (ca.y^ (abyK
Here k^, k^, k-s satisfy the relation
^'1 + A^a + k^ = 'S7 — 1.
The advantage of Stroh's series is that it gives all possible relations between the covariants under discussion. It is, however, generally more convenient to have relations between transvectants than to have them between symbolical products. Thus although series (IX) does not give all possible relations, yet it is frequently the more convenient one to use.
By means of series (VII) and (VIII) we may translate Stroh's series into a relation between transvectants. In fact this is what Stroh himself does. This relation has the disadvantage that the coefficients in it are themselves series.
It is convenient to have a short method of referring to Stroh's series ; we therefore introduce the scheme used by Stroh
/i 72 fa
tti rCft fC^
which is distinguishable from Gordan's scheme by the weight being indicated outside the bracket.
67. The quantics of low order furnish very few examples of covariants containing three symbols concerning which Gordan's series gives incomplete information. We have mentioned (f, H)^ as one such case. The co variant ((/, if, /)'' of the sextic / — where i = (/,/)* — is another. The reader will have no difficulty in proving that
Ex. Prove that the covariant of the binary form ax^^ = bx^^ = Cx^''
66-69] gordan's series 69
can be expressed in terms of covariants of higher grade ; and that the
covariant
{ahf {acf (bcf W^ c^^ vanishes identically.
68. Let the three quantics /i,f-2,f3 be made identical, so that
Then if lu ::j> n, all covariants of weight w and degree n can be expressed linearly in terms of those whose grade is not less than
2w 3 ■
If lu = 3w — 2, it is possible to go a step further ; we may express all these covariants in terms of such as are of grade 2m at least ; since covariants of a single quantic of odd grade may be expressed in terms of covariants of higher even grade.
Hence if w = Sm— 8:if'-n, 8<3, all covariants of degree three and weight w can be expressed linearly in terms of those whose grade is 2m at least.
Similarly if w> n, we have
ei = €-2 = €s = € = w — n,
■S7 = w — 'Se.
Then we may express all covariants of degree 3 and weight w
in terms of those whose grade is not less than —- + €=-- — e.
o o
If € is odd the lowest grade given by this is odd unless ct = Sm — 2,
in which case we see on multiplying (XVIII) by
(aby (bey (cay
that we may express covariants of grade 2m— 1, in terms of
covariants of higher grade. Hence if e is odd, all these covariants
2w may be expressed in terms of those whose grade -^ — — e + 1.
If e is even, then as before we take tv = 3m — S, 8<S, and the minimum grade becomes 2m — e.
69. There is one further point in this matter to be noticed. It has been proved that all covariants which are linear in the coefficients of each of three given quantics and are of given
70 THE ALGEBRA OF INVARIANTS [CH. IV
weight w can be expressed in terms of those whose grade is higher than a certain number. Further it has been shewn that among the covariants actually retained, no linear relations can exist. We passed to covaiiants of a single quantic, and deduced that all covariants of degree 3 and weight iv can be expressed in terms of those whose grade is higher than a given number. Is it possible that amongst the covariants retained here, there may exist other relations which do not appear in the general case ? To answer this question we express the symbolical products as transvectants by means of § 53. The product
(ab)'^ - *■ (be)' ax"'+' " '" bx''^ ~ '" Ca;"» " '
where as before C,„ indicates a covariant whose grade is not less than m.
The covariants retained will be — when w i^ n —
{(fuAYjsr, {(AfsYj.f, ((/3,/)»,/.y
where a ^j; -^ . If ?(; = 37/i — 2, there will be one relation : and if
o
w = Sm there will be two relations between these covariants.
Let us suppose all the quantics to become identical. Then
those covariants for which a is odd vanish. Let us suppose
2'W that amongst the remaining covariants for which a-^ -q- a relation
exists, say
Then using Aronhold's operators which may be written for short (/ ^j , (/2 ^j , (f, ^j we obtain
— since Oi is supposed even.
Hence corresponding to a relation between the covariants for a single quantic, we may deduce a relation between the corre- sponding covariants of three different quantics. The results obtained then, for a single quantic, are as complete as those from which they were deduced ; and no linear relation can exist between the retained covaiiants.
69-72] gordan's series 71
70. The covariant
(«6)^ (bcY (cay aa;''-"-^ b^'''^-'' Cx"~''~''
of the binary form a^^ = b^^ = Cx\ can be expressed in terms of
n covariants whose grade is greater than \, provided that A- ^ ^
and IM + v> -^; unless
\ = ^ = 1/ = - .
The verification of the above important theorem is left to the reader; it is really only a restatement of the theorem of §68.
It should be noticed also that a similar theorem is true when the letters a, b, c do not refer to the same quantic ; it is
{aby- (bey (cay aa;'*'-"-^ bx'"'"^''' c^^-*"-" = S C^+i, provided that fx + v>^.
71. Ex. Prove that the following covariants of f=aj^ vanish identically :
((/, ff\ ff " ^ when n = 4A, 4X - 1, or 4X - 2,
{{fjf\fT when n = 4X + l,
{{fjf^^'\fT~'^ when 7i = 4A + 2.
And shew that no other covariants of degree three vanish except those included in the form
U.ff''^\ff- Stroh.
72. Covariants of degree four. It is the object of the present paragraph to determine the conditions that a covariant of
degree four and grade \ ( where X,:|> - J may be expressible in the form
n n n
2C;,+, + (a6)2(6c)'2(ca)2.^:
the expression C^+i denotes — as before — a covariant of grade not less than X, + 1, and ^ being a covariant of degree one can only be the quantic itself. Of course the second term cannot appear if n is odd.
72 THE ALGEBRA OF INVARIANTS [CH. IV
It has already been proved that any covariant
C = {ahy (bcY (cay a^''-"-^ b^''-^-'^ Ca;"~''~^
where fM + v>-^, is of grade gi-eater than \, unless
Auy covariant obtained by convolution from G is of the same form as C, but the indices of some of the determinant factors are increased ; hence any covariant obtained by convolution from G either is of grade greater than X., or else is the invariant
n n n
(abf {bcf (caf . Any transvectant (G, Fy, where F is any binary form, is also of grade greater than X or else, if p = 0 and \ = fj, = v = -^, has a factor
n n n
(a6)2(6c)2(ca)2.
Further, any term of this transvectant differs from the whole transvectant, by a linear function of transvectants
{c,Fy',
where C and F are obtained by convolution from G and F. Hence any term of a transvectant {G, Fy is equal to
lG,+, + (abf^(bcy(ca)K^,
where as before G^^+i is used to denote a covariant of grade X + 1 at least.
Again, any covariant having a symbolical factor
(abY (bcY (cay,
where /x, + y > ^ , is a term of a transvectant (G, Fy, and hence may
be expressed in the form
SC;,+i + (a6)2(6c)2(ca)2.^.
Consider now the covariant of degree four where v If-X
K = (abf (bey (cdy aa,""^ bx''-^-" c^"-''-'' d^'"".
gordan's series 73
If /i > - , then
K= tC^+, + {ahf {hcf {caf . ^. Otherwise by means of the relation
{cd) ax = (ad) Cx — (ac) dx we obtain, since ^ <t: X <^ i^,
^ = 2(-)<(;)£„ . where
Li = (abf (bey (ac)'^^ {adf aa,"-^-" hx''-^-'^ c^'-i'-'^^ dx"^.
If either
. \
or
. \
then
Li = SC,+, + (ab)hbc)hca)i ^. But if
yu. + 1/ > X,
one of the above inequalities must be satisfied ; hence in this case
n n n
K = 20x+i + {abf (bey (caf . ^.
Next consider the most general symbolical product ^ of degree four ; it is sufficient to write down its determinant factors, which we take to be
(aby (acY' (bcY' (ad)"' (bd)"^ (cd)"*.
It is supposed here that no index is greater than X, which is
n itself not greater than ^ .
By means of the identities
(cd) ax = (ad) Cx — (oa)) dx,
(cd) bx = (bd) Cx — (be) dx,
we can express K in terms of covariants L in which either (i) the index of (cd) is zero, or (ii) the indices of both ax and bx are zero.
74 THE ALGEBRA OF INVARIANTS [CH. IV
The indices for the covariants L will be denoted by accented letters. In the second case
X-\- fj^ -\-Vi =11, \ + fi^' + V2' = n, hence
fh' + fjh+ vi + v^ = 2?? - 2\ ^ 2X,
consequently either
/a/ + /a./ > 2 or i/j' + j/g' > 2 • Therefore
K=XL= SOx+i + {ahY {hcf {cay ^. In the first case
A^l' + tl2 + Vi + I// = /tti + /*2 + ^1 + ^2 + J'S,
and if this sum is greater than \, the same result is true. Hence — Wken fiy + n^-v Vi-\- v^-\- Vz> \
the covariant of degree four ivhose determinant factors are
(abY (acY^ (bcY' (adY' (bdY' (cdY\ may be expressed in the form
XG,+, + {aby(bcf{cay^.
73. Any covariant which contains the symbolical factor
(abYibcYXcdY,
n where /a + j/ > \, and X:lf> ^, may be expressed in the form
tG^+. + iabf'ibcf^cay'^.
For if r be such a covariant, and
K = {abY (bcY (cdY «x''~^ &/"^"'' c,^"-''-'' dj'-" ; then _ _
Each term of this sum has just been proved to be of the required form.
72-76] gordan's series 75
74. Any covariant of the covariant
Ar^2n-2>- = ^ahy a^"-'- 6^"-'-, (7- :|> |^
of the binary qumitic f= Ux^ = bx' = ..., may be expressed in the form
n n n
tC,+, + (ab)HbcY(cay^,
where Cr+\ represents a covariant of grade not less than r-\-\.
To prove this we observe that any form obtained by convolution from a product
{ahy ax''-'' bx''-'' {cdf Cx''-' dx""-"" ■ . .
is either of grade greater than r or else has a factor of the form
(aby{bcy(cdy.
In the latter case by § 73 this covariant may be expressed in the form
71 n V
^Cr+^ + {aby'(hcy{cay.^.
Now any covariant of kx^"~'-^'' may he expressed in terms of transvectants of the form
({k, ky, n.
But the transvectant (k, ky is a linear function of covariants of / obtained by convolution from
{aby ax''-' bx""-'' . {cdy Cx''-'' dJ'-\ Hence the theorem is true for covariants of the second degree ; and therefore for all covariants of kx^'"^''.
75. It is well to notice that nowhere in the last three paragraphs has it been assumed that two different symbolical letters refer to the same quantic. The theorems are thus true when some of the letters refer to different quantics. For example the theorem of § 74 is true for covariants of
lgm-'2r ^ (^jy ^^n-r J^^n-r
when the quantics ax', bx" are different.
76. The discussion of covariants of degree four ma}' be carried on a step further.
Thus if X is even, and
fi + p=\:^-,
76 THE ALGEBRA OF INVARIANTS [CH. IV
then the covariant
K= {ahf {bcf (cd)" a^"- Hj''^-" cj' " '^ " " tf^" " " differs from
2,
by an expression of the form
To prove this we notice that the index of 6^ in ^ is » - X - /x, and this is not less than v ; for by the inequalities above we see that
hence we may use the identity
(cd) hx = {bd) Cx - {be) dx
to obtain
A--2(-1).Qa
where
Now if
. X
• ^
or ii + v-i>-,
then A=SC\+i.
(The weight of ^ is 2X ^« ; hence no term with the factor
n n n {ab)^{bc)^{cayi can appear.)
Then the only term we need consider is
X A _^ _6
L, = {abf{bcfibdfaJ'-'bJ'-'^C''d^'''\
2
Again, let K' be that particular covariant of the form K for which ^=0,v=\. Then
76-77] gordan's series 77
Hence
In just the same way it may be proved that if X is odd, and
(1) f. + v=\:^-,
then K=2C\^^ :
n
or (ii) /x + v + l=X j>-,
then
X- 1 X-1
2
The results of this paragraph may be stated as follows : (i) IfXbe even and fi + v = X, then the covariant {aby {bcY (cdy a.,."-^ 6^;"-^-'" c,."-'^-'' c?.,."-"
can be expressed as a sum of a reducible covariant and covanants
of grade greater than \.
(ii) If \ be odd and fi+ i' = X, then the above covariant can be expressed as a sum of covariants of grade greater than \.
IfXbe odd and /ji + v = X—'l, the covariant can be expressed as a sum of a reducible covariant and covariants of grade greater than X.
Ex. (i). Any covariant which contains the factor
{abf {acp {bc^' {ady (bd)"' {cdY\
where ^i+/x2 + j'i + i'v, + i'3>X and X<-,
can be expressed linearly in terms of covariants whose grade is greater than X.
Ex. (ii). Prove that if a covariant C of degree 5 has the factor written down in the last question, for which
/ii + /H2 + «'i + i'2 + i'3=X, and X:f»-, then C='2,C^,^+ reducible terms.
77. Jacobians. The first transvectant of two binary forms fx^> 4^x^ is called their Jacobian. It is, in fact, equal to
mn d {Xi, Xj.)
78
THE ALGEBRA OF INVARIANTS
[CH. IV
The following properties of Jacobians are important.
(i) If f, (f>, y^ he three binary forms, each of order greater than unity, the Jacobian of the Jacobian of f and <p with y^ is 7'educible.
Let /=tta;^ <f> = b^\ yir^c^P
{f<\>) = {ab)a^^-'h^''-\
Polarize once with respect to y, the result is
^■^' ^^^ = ^/i^^ ^"^^ ''"""' ^^"~' *^ -^ m + n--"2 ^"^^^ '*^"" ^^"" ^^-
Hence
(m + n-2)((/,<^),^/r)
= {m - 1) (a6) {ac) a^""-- b:^"'' c/-^ + (?i - 1) (ab) (be) a^"'-' 6/-- c/"'. But 2 (a6) (ac) ^j^; c^ = (abf c^^ + {acf 6^' - (60)^ a^^^
and 2 (a6) (6c) a^ c^, = - {ahf c^ - {bcf a^' + {acf b^? *.
Therefore
= iaa;'""'6a,'*-''c/- =
m — n w + n — 2
{aby c/ + (acY b^' - (be)- a^'
m — n
-^^^^^_^{f<f>r.ir+hAffr.<f>-H<t>,i^y'f-{^i^)-
(ii) The product of two Jacobians may be expressed as a sum of products of covariants, ther^e being at least three covariants in each product; provided the forms of luhich the Jacobians are taken are all of order greater than unity.
To prove this, we first establish a useful identity between symbolical forms.
Consider the determinant
|
(/,- |
Wl«2 |
ai |
|
b{^ |
bA |
b,' |
|
c,' |
C1C2 |
Cf |
til «9
it vanishes if -p = r . hence (ab) is a factor ; similarly [be), (ca) are
factors. There can only be besides a numerical factor, which may
* See Chapter I. § 22.
77]
GORDAN S SERIES
79
be determined by considering the coefficient of a^hih^c^. The determinant is therefore equal to
— {ah) (be) (ca).
Hence 2 . (ab) (be) (ca) . (de) (ef) (fd)
di - 2d,d. di" 62' Z6162 61
fi -2/,/, y,^
(adj (aef (aff
(bdy (bef (b/y (XX).
(cdf (cey (c/y \
In this identity let us put
Ci == sCo, C2 = ^1 5 y 1 ^ '^■2> j2^ ^i-
|
a^' |
a^a^ |
ai |
|
|
w |
bA |
bi |
|
|
c,' |
C1C2 |
ci |
Then
2 (ab) ax b^ . (de) dx e^ =
(ady (aey Ox^
(bdy (bey bx'
dx' e^ 0
Consider now two Jacobians (f, (j)), (yjr, -)() ; where /=a^- 0 = 6^",
Then
(/ 4>) ■ (^. %) = (a^) ax'""-' ^x'-' . (de) dx^-' e^'"^
= \ax'^-'bx''-'dx^-'ex'i-'
(ady (aey a^
(bdy (bey bx'
dx- ex- 0
+ i(<^.t)yX-i(0.%)V-f (XXI).
For the sake of generality the forms /, <^, i/r, ^ have been supposed to be all different. The theorem is still true if two are equal to one another. In particular, it is true for the square of a Jacobian.
80
THE ALGEBRA OF INVARIANTS
[CH. IV
Another symbolical identity of interest is obtained thu-s : Form by means of the ordinary law of multiplication the product of the following vanishing determinants :
|
ajttg |
ai |
0 |
|
|
Wh |
V |
0 |
|
|
CiC.2 |
«•/ |
0 |
|
|
d^d.^ |
di |
0 |
Se^ea
0
fi -2/,/2 f^ 0 9% -'^9x9% 9i 0
{agf (ahy (Jbgf {bhf {cgf {chf
= 0 (XXII).
Hence if
= 0.
{aef {a/y
{bey {hfy
{cef {c/y
{def {dff {dgY (dh)
F=e^'', *=//', ■9=g/, X=A/
i (/, Ff (/, *)2 (/, •^f (/, x)2
^ {<i>,Ff ((^,*)2 (0,<?)2 (0,x)2
(v^.i^)-'' (>/.,*)i' (>/.,*)2 {^,xy
U,Ff (;^, *)2 (;f,*)2 (;t, X)^
Here, as before, the forms are not necessarily all diflFerent.
78. The expression for the product of two Jacobians may also be obtained as follows.
We have
(/ «^) X i-^, X) = («^) dx'^-'ho,''-' • {de) d,P-'e,9-'
= (ab) (ae) a/^-^ft^^-^e^?-! . ^jr - {ah) {ad) a^'^-^b^^'-^d^P-' . x- and, by means of the identities of the type
{ab) {ae) b^e^ = {ab^ e^ + (ae)'- bi - {hef ai, the right-hand side becomes
\ (/ xr <f>f + ^ i<f>> n/x - i (/ ^)^ <t>x-i (4>, xT-ff
as before.
Thus if J be the Jacobian of y and 0
- 2J^ = {fjy <f>^- + (</,, cf>yf^ - 2 (/, <f>)\f<i>.
79. Copied forms. If in the symbolical expression of a covariant 11 of a binary form F^^, the symbolical letters are taken to refer to another binary form <f>x^ where m^p, and 11 is multi- plied by
aJ''-PbJ"-v ...
77-79] gordan's series 81
— where a,b, ... are the letters occurring in H — then the resulting form IT' is called a copied form. The original form 11 is called the model form.
Ex. The Hessican of any binary form (l> = a^^^ = hx^^ is (a6)2a;,'»-26;,'»-2. It is formed on the model of the Hessian of the quadratic.
The use which we are going to make of the idea of copied forms is for the case when the orders p and m of the fundamental quantics of the two systems are the same. The form ^ will be taken to be a covariant of degree two and order m of a binary form /x'*: then
<^ = (/, ff<^ = {ahf" a^""--" ba,''--" = (f>x^-*^, 2n-4ia = m.
On the model of the complete system for the general binary form F^^, we may construct a complete system for (f). If when this is done the symbolical letters <^i, <f)2, ... which refer to (f) are replaced by the symbolical letters a, h, ... which refer to yj each covariant of </> will become a covariant of/, which consists of a sum of sj'mbolical products instead of a single term. These separate products may be arranged in a series of adjacent terms, the difference between any two of which contains a factor of the foim (a6)^'^"'"'. Thus if we reject covariants of grade greater than 1(T, any one of the terms given by a covariant of ^ may be taken to represent this covariant. Before proceeding to a rigid proof of this statement, let us consider an example.
The covariant of the sextic
"z — "x — • • •
is a quartic. Let us consider the Hessian of <^,
It consists of a linear function of ten covariants of the following tj'pes :
{abY{cdf{acfb^^d^\
{ahy{cdY{ac){ad)h:?c^d^,
{aby (cd)* (ac) (bd) a^bxCxd^.
Any one of these may be taken for the copied form, for each differs from the whole transvectant by transvectants of forms obtained from
iab)*ax^bx% [cdfcx^d^^ by convolution. Hence when forms of grade higher than 4 are neglected, the whole transvectant and each of its terms are equivalent.
G. & Y. 6
82 THE ALGEBRA OF INVARIANTS [CH. IV
80. Consider any symbolical product P, the symbols of which refer to
in this product let us replace any particular letter <^<^' by a new variable y ; (we replace ^12 by y^ and ^u by — ^o). Let Py^n-w be the resulting expression, then
= (Pj,2n-4<r , {aVf^ ay''-'"' by"-'"'y"-^ .
Any term of this transvectant dififers from the whole trans- vectant by terms involving a factor (aft)"^"'"'"' : for since y is absent when the transvectant is expanded, the only kind of adjacent terms are of the form
(aa) (y86) M, (ab) (/3a) M (see § 50).
The above argument is not affected if we suppose that symbolical letters are present in P which do not refer to <f>. Hence we may replace each of the letters which refer to (f> in turn by letters which refer to /; and in doing this we may at each stage choose any one term to represent the whole expression P, provided that those terms which involve (aft)-""*"' are rejected. Thus taking the quartic invariant {abf(bcy(cay for model, we obtain the copied invariant of
<\) = {abya^^b^\
viz. {(f>M^ i^ifk^iy ((t>3<f>iY-
Any one of the sextic invariants
(aby(cdy{efy(acy(dfr{eby,
{aby(cdy{efy{ady{cey(fby,
(aby (cdy (e/y (ac) (ad) (ce) (de) {/by
differs from the invariant of (f>, by terms involving the factor {abf.
Conversely in any co variant of a binary form aa;"= 6/'= ..., which has a symbolical factor (ab)-'', we may replace the letters a, b wherever they occur in the symbolical product by a single letter <f>, and remove the factor (aby'^ altogether: on the under- standing that forms of grade greater than 2a are being rejected, and that 0 refers to the form (a6)-'^aa;"~-'^6a;"~-^ The reader will have no difficulty in verifying this statement.
80-82] gordan's series 83
81. G-eneralized Transvectants. Consider a product
of any binary forms,
/i(^) = ax'", My) = hy'\ fM = o^P.
The result of operating on P with
{m-\-^l)\ {n-\-v)\ (p-^^iiOUx Of^ O^ m\ ■ n\ 'pi ^^^y^^^z^^z,,
where
n.,= ^ - ^
*■' dx-idy^ dx^di/i '
after operation y and z being replaced by a;, is
(aby {acY (chy aa;"*-^-'*6a;"-■^-''Ca;^-'*"^
This we define to be a generalized transvectant.
Instead of taking forms each having a single symbolical letter, we may construct generalized transvectants of forms each of which has two or more symbolical letters. Thus we may replace a^;'" by
The generalized transvectant may then be expanded as a linear function of certain symbolical products. Just as in the case of ordinary transvectants, any term of a generalized transvectant differs from the whole transvectant by lower transvectants of forms obtained from the original forms by convolution. We leave the verification of this statement to the reader.
Further it is evident that any symbolical product may be regarded as a generalized transvectant. A copied form is then merely the same generalized transvectant with a new form taken for ground form.
The theorem of § 80 — that any single term of a copied form, when a covariant of the second degree is the new ground form, may be taken to represent the whole form, provided that forms of higher grade than that of the fundamental ground form are to be neglected — is merely a particular case of that just enunciated.
82. Hyperdeterminants. When the forms of a generalized ti-ansvectant are all the same, it will be noticed that the trans-
6—2
84 THE ALGEBRA OF INVARIANTS [CH. IV
vectant is entirely given by the differential operator. Thus a covariant of the binary form /= ax^, is completely defined by the operator
^^ xtj i^ xz ^''"zy
Cayley used the notation
to define such a covariant. These symbolical forms are called hyperdeterminants. Cayley introduced his calculus of hyper- determinants some years before the symbolical notation was invented by Aronhold.
The hyperdeterminant notation was introduced for a single binary form merely for convenience. It is evident that it may be used perfectly well for covariants of two or more different forms.
It is interesting to notice that the letters of a symbolical product may be regarded as differential operators. Thus if
_ a p _ a_ _d
"'-d^,' ^^"a.7i' '^^~afi'
_ a „ 8 d
dfa 07]^ 0?2
then {a^Y (ajY (7/3)"
operating on the product of
produces the covariant
{ahY {acY (chY tta:"*"^"'* &«""*"" Cx^*""" multiplied by
ml 111 p\
(m — \ — fi) \ {n — \ — v) I {p — fj, — v)\'
provided that after operation ^, rj, ^ are each replaced by x. Further the operator
(a^Y (<^yY (j^y olT-^-'^ /S^"-^-" 7xP-'^-'' acting on the same product produces the same covariant multi- plied by
m\n\p\.
In this case ^, 17, ^ all disappear after operation, so there is no question of replacing them by z.
CHAPTER V.
ELEMENTARY COMPLETE SYSTEMS.
83. Complete Systems of irreducible covariants. We
shall devote this chapter to a detailed discussion of the invariants and covariants of single binary forms of the first four orders ; in particular, we shall obtain what are known as the complete systems of covariants for such forms. It has been observed, in fact, that the symbolical notation enables us to construct an infinite number of covariants of any form y*, but, as was first proved by Gordan, all these are rational integral functions of a finite number of covariants of f\ this finite number is said to constitute the complete system of irreducible concomitants, or, more briefly, the complete system of concomitants of the form. The general proof of Gordan's Theorem will be given in the next chapter. For the present we shall content ourselves with explaining easier methods of obtaining the complete systems in the simpler cases, and proving of course that such systems are actually complete.
Inasmuch as every covariant can be expressed as an aggregate of symbolical products, we need only retain such as consist of one product in seeking for the complete system.
84. Linear Form. The discussion of a single linear form
/=a* = &x = etc. presents no difficulty.
For a symbolical product either contains a factor of the type (a6) or it does not. If it does so it is zero because
(a6) = 0
and if it does not it is simply a power of/.
86 THE ALGEBRA OF INVARIANTS [CH. V
Hence every covariant of a linear form is a power of the form itself, or in other words, the form constitutes the complete system.
Go7\ The same argument applies to any number of linear forms, for every symbolical product is a rational integral function of invariants of the type (ab) and co variants of the type ag.. Hence the complete system for n linear forms consists of the n forms themselves and the ^n{n — l) non-vanishing invariants of the type (ab).
This result has already been established in § 37 where it forms the lemma preliminary to the proof of the fundamental theorem.
85. Quadratic Form. Suppose the form is
/= cix^ = bx^ = Cx^ = etc.
Then if a symbolical product contain no factor of the type {ab) it is a power of / ; if on the contrary it contains such a factor {ah) it can be transformed so as to contain {ab)- (§ 60), which is an invariant. Thus every invariant and covariant except
^x, {ahy
can be expressed in terms of co variants of lower degree, hence by continued reduction we infer that every such form is a rational integral function of
/(/./)- A,
the latter being the discriminant of the quadratic. In other words, the form and its discriminant constitute the complete system.
Ex. Prove that
{ah) (ac) {bd) (ce) dxOx = jAy.
(By interchanging a and h put the factor {abY in evidence.)
86. Before proceeding to the discussion of the cubic and the quartic we shall explain some general principles relating to the formation of the irreducible co variants of any given degree of a binary form.
Suppose that the form in question is
/= tta," = 6^" = c^'' = etc.
then the only irreducible covariant of degree one is/
■ 84-cSfil
84-86] ELEMENTARY COMPLETE SYSTEMS 87
Next, the only covariants of degree two are those of the type
(a6)'-a^'»-'-6^''^-''; r = 0, 1, 2, ... 7i.
If r be odd this covariant vanishes, and if r be zero it is reducible since it is equal to f^; the remaining forms corre- sponding to even values of r constitute the complete set of irreducible covariants of the second degree.
Now assuming a knowledge of all the irreducible covariants of degree less than m we shall shew how to find the irreducible covariants of degree m.
Suppose that the given irreducible forms are
then any covariant of degree less than m is a rational integral function of / and the ^'s. Now a covariant of degree m is an aggregate of symbolical products containing m letters ; let C^ be one of the products and k one of the symbols involved, then Cm is a term in a transvectant
where G-„i-\ is a product containing only ni — 1 letters, that is, it is a covariant of degree m — 1.
Thus 6',, = (6',„_: , fy + S (C^, /)"', p'<p
and C',„_i is derived fi-om Cni-\ by convolution.
But C„,_j, C,rt_i being covariants of degree m— 1 are rational integral functions of the forms
/ 01, 0i, ••• 0r,
i.e. they are aggregates of terms of the types
f^m-i=/''0,'^' ... 0A
of degree m — \.
Consequently (7,„ is a sum of transvectants of the form
(f^^n-i,/)",
where of course p :|>- n. Since this is true for every separate term in a covariant of degree m it is true for the whole ; or, in other Avords, every covariant of degree m is expressible in terms of transvectants
88 THE ALGEBRA OF INVARIANTS [CH. V
where Um-i is a product of the form
and is of degree m — 1.
Hence to find all the irreducible covariants of degree m we have to write down all transvectants of the form
and reduce as many of them as possible. The remaining ones are the irreducible forms of degree m, for any covariant of degree m can be expressed in terms of them and covariants of lower degree.
For example, in the case of a binary quintic the irreducible forms of degrees one and two are
f,H= (aby a^ b^, i = {aby a^ b^.
The only products of powers of these of degree two are f"^, H and i, so that all the irreducible covariants of degree three are included in
(/^/)^ {Hjy, {ijy,
where p ::|» 5 for the first two transvectants and p :^ 2 for the third, since i is a quadratic.
87. Let us now return to the transvectants
{'Urn-.fy
which we have to reduce as far as possible. That many of them are reducible follows from the following principles.
Suppose that ^m-i = yW
where Fand W are likewise products of powers of/, <^i, i/),, ... <^,., but of smaller degree than Um-\, and further suppose the order of W is not less than p.
Then if T^ be any term belonging to V and Tn_ any term of the transvectant {W,fy, which is a possible transvectant because the order of W is at least equal to p, T^T., will be a term in the transvectant
(Um-^jy.
Hence
{U,n-^,fy=T,T,+ t{[I,r^„fy■,p'<p
and Ti, T^ being both covariants of degree less than m are expressible in terms of f and the ^'s.
86-88] ELEMENTARY COMPLETE SYSTEMS 89
Now in discussing the reducibility of the transvectants
let us consider them in the order of their indices, e.g. we examine all those of index one before we proceed to any of index two, and so on.
Then since Um-\ is a covariant of degree m — 1 it is an aggregate of products of the type Um-\ , and since p < p it follows from the equation
that the transvectant on the left is completely expressible in terms of co variants of less degree and transvectants previously considered, or to put the matter briefly, it is reducible for it certainly cannot give rise to a new irreducible form.
Hence the irreducible covariants of degree m can only arise from such transvectants
{Um-^,fy
for which Um-i has not a factor of order greater than p. Thus in the case of the quintic no transvectant of the type
(A/y
is irreducible because p 1f> 5 and the term f^ contains a factor / whose order is 5.
In the general case if Um-i possess a factor W whose order is not less than n, then the product Um-i can give rise to irreducible covariants for no value of p ; it may therefore be neglected entirely in the search for irreducible covariants.
As an application of this remark we note that if the order of one of the </)'s, say (f>p, be at least equal to n, then we need not consider the product of this form with any others.
Finally if 0g be an invariant we may leave it out of account in forming the transvectants, because it would occur as a factor in each transvectant in which it appeared and so the transvectant would be reducible.
88. Irreducible system for the binary cubic. After these preliminary explanations the deduction of the complete system of a cubic presents little difficulty.
90 THE ALGEBRA OF INVARIANTS [CH. V
Let the form be
f=a^^^ hi = d = etc.
then the only irreducible form of degree two is
H={ahya^K = hi = h'i.
To find the irreducible forms of degree three we note that the product /- is negligeable, so the only possible irreducible forms are
{H,f), {Hjy.
Now
(H,f) = {ahy{ac)hci = -t, where t is an irreducible covariant, and
{H, fy = {{aby a^ h, c^]' = (aby (ac) (be) c^ = — (be) (ca) (ab) {(ab) Cx} = — ^ (be) (ca) (ab) {(ah) c^ + (be) a^ + (ab) Cx], as we see by interchanging a, c and b, c and adding the results. Hence (H,/)'- = 0 and the only new irreducible form is t. The prodvicts of degree three are
/'. a/, t,
of which the first two may be neglected ; hence the irreducible forms of degree four are included in
(tj), (t,f)\ (tjy.
Of these (t,f) is the Jacobian of a Jacobian and hence can be expressed in terms of forms of lower degree, § 77.
In fact, to give the actual expression, we have {(H,f), ^] = i (/, ^fH-\ (H, iryf, whatever "^/r may be, since (H, fy = 0. Therefore - (tj) = i (fjy H-\ (Hjyf,
or (t,f) = -\H\
Next
(t,fy=^-\(ha)hxai,bx'Y
= - (ha) (hb) (ab) axbx + X [(hay ax, 6*^}, since (ha) (hb) (ab) a^ bx
is one term in the transvectant.
88-89] ELEMENTARY COMPLETE SYSTEMS 91
Now this term vanishes, and since
we have
Finally
(t, ff = - {{abf (ac) h o^\ d^f = - {ahj (ac) (bd) {cdf = - A, so that (/, tf = A an invariant which proves to be irreducible. The products formed from/I H, t which are of degree four are /, f^H, ft, H\
and all except H- may be rejected because t and / are both of order three.
Further {H^,fy is reducible unless p > 2 because H- contains the factor H whose order is two.
Hence the only possible irreducible form of degree five is {H'\ff. But
{H\fy = {K'h'^\ a^J = {haf{h'a)h'^
= — [{hd)- ax, h'a;-] = 0 since (ha)'- a^ = 0,
hence there are no irreducible forms of degree five, and in fact it is easy to see that there are no more irreducible forms. For if there were, the one coming next in ascending degree would be of the form
(f^HHyjy.
The only products that can lead to irreducible forms are f, t, H and H-, because when /3 > 2, H^ involves the factor H- whose order is greater than three ; but the transvectants arising from each of these products have already been considered, hence there are no more irreducible forms ; in other words, every invariant or covariant of the cubic is a rational integral function of / H, t and A.
89. Irreducible system for the quartic. If the form be
J = Qjx =^ Ox = Ca- ,
then the irreducible forms of degree two are
H = (abyax^hx- and i^{ab)*, H being of order four and i an invariant.
92 THE ALGEBRA OF INVARIANTS [CH. V
The only product of degree two that we need consider is H, and hence the irreducible forms of degree three are included in
{Hjf, (Hjy, (Hjy, {Hjy.
Now
(S>fy = W (ac) a^bJ'Ca,^ = - t where t is an irreducible covariant.
{H,fy={{abyaJ^K\c,^Y = j\ (aby (acy bjc^' + j\ (aby (ac) (be) a^b^c^^ + ^ (aby (bey a^W = i (aby (acy ba^'c^' + f (aby (ac) (be) a^b^c^^ = i (aby (acy b^^c^^ + \ (aby c^^ [(acy b^' + (bey a/ - (aby c^^] = (aby(acyb^'cJ'-Uahy.Ca?, since the symbols are all equivalent. Then since, § 22, (aby c^' + (bey a^' + (cay b^*
= 2 (aby (oAiy ba?e^' + 2 (bay (bey a^^d + 2 (cay (cby a^b^, we have (aby (acy b^c^' = \ (abf . e/.
Therefore (H,fy = W- W= ¥f
and is reducible.
Next (Hjy = {(aby aj'b^', e^'Y = («^)' i^c)' W b^
for the two terms in the traiisvectant are equivalent. By inter- changing a and c it follows that (H,fy = 0.
Finally
(H,fy = {(aby a A', Ca^'Y = io,by (bey (cay =j
an invariant ; hence the only irreducible forms of degree three are t and j.
The only product /"H^ty of degree three that we need consider is t, and the only possible irreducible forms of degree four are therefore
(tjy, (tjy, (tjy, (tjy.
As a matter of fact these are all reducible.
89] ELEMENTARY COMPLETE SYSTEMS 93
To calculate them we use the series of ^ 54, with the scheme
f H f
4 4 4 0 r 1
which gives
or
'4 — r\ /I
/4 - rW
=2^v^^i(/,/)-^)-'.
If r = 1 we find
since (f, Hy = ^if,
hence (t,f) = ^H"-j\if'.
Ifr=2,
(t,fy + {{f,Hyj] + j'A(fHfJY
and since (f,HY = ^if, {fHf = 0,
this gives (^./)' = 0-
From r = 3,
(t, fy + f {(/> sy-jY + A {(/ ^)^/} + i (/ ^)^ •/
= Ufy,H} + ^(/,/y.H,
or a/>'^-i^•5^ + lJ/=i^i^,
on putting in the values for the transvectants off and H.
Thus
{t,fy=i{iH-jf).
The series does not apply when r = 4 because then r + 1 > 4, but it is easy to calculate
(tjy
directly or as in § 94.
94 THE ALGEBRA OF INVARIANTS [CH. V
For
{(ah) a^'h^\ K*Y = («^0 («&)' ^* V + ^ \{ahy a^%^\ b^'Y
+ fji {(ahy a^K, h^*Y + ^ {i<^hy, b^'}, while (aby {ah) b^^h^^ = [{aby a^b^, V} = 0,
{ahya^'h^' = iif; (/,/>' = 0, {ahy a^hx = 0, hence (^,/)* = 0.
Hence there are no irreducible forms of degree four. In fact there are no more irreducible forms, because the next in order of degree would be of the form
(/"HHy,/)".
Now since /, H, t are each of order four at least, irreducible forms can only arise from products containing each of the three by itself, and all these, viz.
ifjy, (sjy, (t,fy
have been already considered. Hence every invariant or covariant of the quartic is a rational integral function of/, H, t, i and j.
90. Quintic. To illustrate still further the method of this chapter we shall apply it to some extent to the binary quintic. The covariants of degree two are
(/,//= ZT and {f,fy = i.
The products of powers of /, H and i which are of degree two are /^ H and i, and to find the covariants of degree three we have to consider transvectants of these three forms with f.
The transvectants arising from /-' may be neglected, and hence we are left with
{H,f)\ (H,/)\ (Hjy, {H,f)\ (Hjy,
{i,f)\ (i/y
as the only possible irreducible covariants of degree three. Now of these
(H,fy = {{aby a^b^\ c^'^Y
89-90] ELEMENTARY COMPLETE SYSTEMS 95
and involves a term ^
{aby (ac)' a^bai'c^',
.-. (Hjy = (aby (acy a^b^^c^^ + \ {(aby a^b^, c^»|«
= ^axKcx {{ciby (acy b^Cx' + (6af (6c)- c^^ai
+ {cay (cby ax%^] + \if = ^a^hcx I {(aby Cx* + (bey a^' + (cay 6^^} + \if = (\ + ^)if, so that (H,fy is reducible.
Again (H, fy = \(aby a^b^, c^
and contains the term
(aby(acybxHx\
This terra can be transformed so as to contain (ocV and hence must be a multiple of
(aby (ac) b^c^*
since the letters are equivalent.
.-. (H,fy = X (aby (ac) b^Cx* + fi {(aby a^bx, c^»|
= (\ + fM)(i,f).
Further
(H, fy = (aby (bey (acy a^b^c^ + \ a fy
and
(H,fy = (aby(bcy(acya,
= i (bey (cay (aby {(bo) a^ + (ca) b^ + (ab) c^] = 0. Finally (i,fy is an irreducible form and (i,fy = (aby(ac)(bc)c,^
= - i (be) (ca) (ab) {(aby c^' + (bey a^' + (cay b^'} = - (bey (cay (aby a^b^c^ (§ 22).
Hence the only irreducible covariants of degree three are
(H,f) = t,(f,i)
and (/, iy = - (bey (cay (aby a^b^c^ = -j.
The reader may now hnd the irreducible forms of degree four and verify the result by reference to the chapter on the quintic. It will be seen at once that the method leads to much labour, that the reduction processes are not easy to discover, ai^d, when we mention that for the quintic we have to proceed step by step
96 THE ALGEBRA OF INVARIANTS [CH. V
until we get to degree 18 before the irreducible system is obtained, the impracticability of these methods in dealing with forms of order greater than four will be at once admitted.
91. Further Theory of the Cubic. Syzygy among the irreducible forms. There is an identical relation con- necting the irreducible concomitants of the binary cubic — the simplest example of what is known as a syzygy among the covariants of a single binary form.
In fact since t is the Jacobian of/ and H we have, § 78,
- 2t^ = (f,ffH^ + {H, Hyp-^iHJfHf. Now
{H,ff=^{ahfa^ = 0
(H, Hy = [{ahf a^h, {cdf cM' = W (c^)' («c) (bd) = A,
for although there are four terms in the transvectant they are identical in value, and we have
the relation required.
92. Since every covariant of the cubic is a rational integi-al function of/, H, t and A it follows that all expressions derived by convolution from products of powers of/ JET and t can be expressed in terms of/ H, t and A.
The form of the expression can be easily inferred by con- sideration of its degree and order, but the actual determination of the coefficients may be a troublesome process.
As an example consider the Hessian of t, i.e. {t, tf. It is a covariant of degree six and order two since t is of degree three, and the only product of / H, t, A fulfilling these conditions is ^A. We at once see that (t, tf is a numerical multiple of ZTA.
The reader may calculate the actual value directly by using the series of § 54 — we give an alternative process.
Let J=(t,f),
then - 2J^ = {t, typ + if, ff P - 2 (/ tyft
90-93] ELEMENTARY COMPLETE SYSTEMS 97
But {t,f) = ^H\ {f,fy=H, (f,ty=o
and t' = -\H^-l Ap ;
therefore -^H' = (t tfp ^H{-\E'-\ A/^) ;
that is {t,tfp=\ApH
or {t,ty=lAH.
Again, consider the symbolical product {ahy (ac) (bd) (cd) Cxdx which represents a covariant of degree four and order two.
Since there is no product
of this degree and order the covariant in question must vanish identically.
To verify this we remark that, on interchanging a, b and c, d, the expression
(aby (ac) (bd) (cd) c^d^
changes sign ; hence it vanishes.
Ex. (i). Calculate the following transvectants in terms of/, II, t, A, viz.
{ff, H), {H, H)% it, t), {t, t)\ {t, tf, (IT, t), {H, tf.
(The only one presenting any difficulty is {H, t) and this is the Jacobian of a Jacobian ; its value is — ^ A/.)
Ex. (ii). Shew that any symbolical product involving the factor {aKf vanishes identically.
Ex. (iii). Shew that
{m,pf = 0, {H^fif = 0, {H\P)=^H'^f{HJ)=-H^ft.
93. Further Theory of the Quartic. As in the case of the cubic, the square of the covariant t can be expressed rationally in terms of the remaining forms.
In fact we have, § 78,
- 2t^ = (f,ffH^-2(H,fyHf+(H, Hfp,
while {f,fr = H, (HJf = iif,
so that it only remains to calculate (H, H)'\
G. & Y. 7
98 THE ALGEBRA OF INVARIANTS [CH. V
Now using the series of | 54 with the scheme
( 4, 4, 4 J \ 0, 2, 2 /
we have
(') (') (') (')
'f7') 1'7''
or
{H,Hr+[{f,f)\H]^^{fjy.H
= {(/ ^)^/}^ + ((/, Hr,f\ + 1 (/ Hy.f,
that is (£r, Hy + liH = ii (/ /)^ + ^ j/,
since (/, H)'^ = 0 etc.
Hence {H, Hf = iJf-^iH.
Consequently
which is the syzygy required.
94. We shall illustrate the reduction of covariants of the quartic by calculating the values of the transvectants of
/, H, t, taken two together.
The transvectants of f with itself, H and t have already been found.
As regards the trans vectant
(H, Hf
we remark that it vanishes when r is odd and
(H,Hy = yf-iiH.
There only remains (H, Hy.
This is equal to {(ahy a^'h\ h^*]* = (ahy {ahy (bhy = {{ahy a^'h^\ hJ'Y
= [{H,fy,fY=ii{f,fy = li\
93-94] ELEMENTARY COMPLETE SYSTEMS 99
To calculate the transvectants
{t,Hr, r>>3,
apply the series of § o4 with the scheme
/ H, f, H^
( 4, 4, 4 )
\ 0, r, 1 J
and we have
?)(i),„ ..„,„,. .^r
I
On taking r=l, 2, 3 successively and putting in the values of the transvectants (H, HY'^^ we find
(t,H)=if{iH-jf)
{t, Hy = 0
For {t, Hy, the scheme
/ H, f, H^
4, 4, 4 V 1, 3, 1 /
must be used ; or else as in the case of (t,fy it is easy to see that
{t,Hy = o.
To find (t, ty we apply the series with the scheme
{ 4, 4, 6 )
\ 0, 2, 1 / which gives
{(/ H), tY + {{/, Hy, t] + j\ if, Hy t = u ty, h] + § (/ ty h or {t,ty + \{f,t) = l{f,ty.H.
On substituting the values for {t,f) and {t,fy we find
(t,ty = ujfff-^^H^'-^iT)'
7—2
100 THE ALGEBRA OF INVARIANTS [CH. V
For {t, ty we use the scheme
//' ^' «\ I 4, 4. 6 )
\ 1, 3, 1 /
leading to
{t,ty+l(f,ty={{t,fy,BY
-\is,Hy-iif,Hy
and hence {t, ty = 0.
Finally
(t, ty = {{t, fy, HY = i i (H, Hy - ij (/ Hy
Ex. (i). Deduce the value of {t, tY from the relation
-2{(«,/)}2=(«, typ+(f,fyt'-2(f, tfft.
Ex. (ii). Apply Gordau's series to calculate {t, t)^ for the cubic. Ex. (iii). Prove that
Ex. (iv). Prove that for the quartic
{{ff,iif,H}=+yt
{{H,H)\HY = \f-^z\
Ex. (v). Prove that the Hessian of the Hessian of the Hessian of a quartic / is
Ex. (vi). Calculate the values of {{t, t)'^, ty for r=l, 2, 3, 4, 5, 6.
Ex. (vii). If a quartic / be the product of a cubic by one of the linear factors of its Hessian, then
(/,/)*=o.
CHAPTEE VI.
GORDAN'S THEOREM.
95. We have already referred to Gordan's theorem which asserts the existence of a finite complete system of covariants for any binary form, and, in fact, we have illustrated the truth of the theorem in obtaining the complete systems for the quadratic, cubic, and quartic. Our previous method is of little practical utility in dealing with forms of order greater than four, but a comparison between it and the procedure of Gordan may not be without value as a primary indication of the salient features of the latter. In the last chapter covariants were classified according to their degree and we shewed how to obtain those of degree m by transvection from those of less degree. In Gordan's investigation covariants are classified according to their grade — the grade being a definite even number associated with any symbolical product, § 61, and all covariants of grade 2r are obtained by transvection from those of inferior grade together with some of grade 2r.
The advantage of using the grade is that no covariant can be of grade greater than n ; accordingly, the number of steps in the process is small, whereas there being no limit, cb priori, to the degree of an irreducible covariant, and the actual degree reached by irreducible forms of quintics, etc., being very high, the number of steps in the other process is uncertain and at the best large. As will be seen later, on the other hand, the transition from grade 2r — 2 to grade 2r is commonly much more difiicult than that from one degree to the next higher.
Several preliminary propositions are necessary before we can undertake the actual proof of the existence of the complete system ; these we now proceed to explain.
102 THE ALGEBRA OF INVARIANTS [CH. VI
96. The first lemma required belongs to that branch of the theory of numbers known as Diophantine Equations.
For the sake of clearness we shall begin by giving an illustration.
Consider the homogeneous linear equation
2x + 5y = 2z ;
it is easy to see that the number of solutions in positive integers is infinite.
Moreover, if
and so=p\ y = q, z = r\
be two solutions, then
x = 'p-\-'p, y = q + q', z = r + r
is also a solution.
We shall call this latter the sum of the former two solutions, and when any solution can be written as the sum of two smaller solutions (throughout we deal only with solutions in positive integers), it is said to be reducible. Otherwise a solution is irreducible, and the important fact for us is that the number of irreducible solutions is always finite.
Thus for the equation above the only irreducible solutions are
x = S, y = 0, z = 2; a; = 0, y = S, z = 5;
a;=l, y = 2, 2^ = 4; x=2, y = l, z = S.
In fact if a; > 3, then z>2, and the solution can be reduced by means of a; = 3, y = 0, ^ = 2 ; whereas if y > 3, then z>5, and the solution can be reduced by means of « = 0, 2/ = 3, ^= 5 ; thus in an irreducible solution neither x nor y can exceed 3, and, as the number of remaining possibilities is finite, the irreducible solutions can be easily found by trial.
By continually reducing a given solution, say a; =p, y = q,z = r, we can express it in terms of the irreducible solutions, that is in the form
p = SX + v + 2p )
q = Sfi + 2v + p i (A),
r=2\ + 5fi + 4iv + Sp)
where \, fj,, v, p are positive integers :
r
96-97] gordan's theorem 103
e.g. take the solution
x = oO, y = 7, z = 45,
reducing by means of
a; =3, 2/ = 0, z = %
a;=16.3 + 2, 2/ = 7, ^=16.2 + 13; then reducing
x' = 2, y' = 7, 0' = 13
by means of a; = 0, 2/ = 3, 2^ = 5,
a;' = 2, y = 2 . 3 + 1, / = 2 . 5 + 3,
and the remaining part
a^' = % 2/"=l. /' = 3 is irreducible.
Hence in this case we have
\=16, fi=^1, v = 0, p = l.
Of course if we substitute the expressions in (A) for x, y, z the equation is satisfied identically ; the important point is that evei^y positive integral solution can be written in the form there indicated.
97. The idea of reducibility can be extended at once to any number of linear homogeneous equations, for the sum of two solutions is always a solution, and we may enunciate our first lemma as follows :
The number of irreducible solutions in positive integers of a system of homogeneous linear equations is finite.
Consider first a single equation,
a^Xi + a.x^ + . . . + amXm = &i 3/1 + ^2^2 + . • • + hnyn,
connecting the x'b and y^, where the coefficients a, h are positive integers.
If the two solutions
^i^^Si) '^2^62' ••• '^m~ ^m> yi^^Vi' y-2^V2t ••• yn^^Vn'} ^1 = ^1, ^2 = ^2, . • • ^m = ^m, 2/1 = Vi, 1/2 = V2, • • . yn = Vn,
be typified by
104 THE ALGEBRA OF INVARIANTS [CH. VI
respectively, then
also typifies a solution and this latter is reducible. First the equation has mn solutions of the type
with the rest of the variables zero.
Next suppose that in a solution one of the xs (say x-^ is greater than
61 + 62+ •■• +&«,
then the right-hand side of the equation must be greater than
ai(&i + &2+-.. + 6n),
i.e. 61 (2/1 - Oi) + 62 (2/2 - ai) + . • . + hn (y« - aO > 0,
so that at least one y must be greater than a^. Let yr>(h, then the solution in question is reducible by means of the solution
with the other variables zero.
Hence (61 + 62 + . . • + 6„) is an upper limit to the value of any X in an irreducible solution,. similarly (oi + ^2 + • • • + ^m) is an upper limit to the value of any y ; but the number of solutions reducible or irreducible subject to these restrictions is manifestly finite, therefore d fortiori the number of irreducible solutions is finite.
If the irreducible solutions be typified by
«? = «!, 2/ = /3i ; sc = (h, y = ^2]
« = «P' y = ^i»
then by continued reduction any solution can be expressed in the typical form
a; = ^iffi + ^2«2 + . • • + t^ap
y = t,^, + t^^,+ ...+t,/3„
where the t's are all positive integers.
Suppose now we have a second equation of the same nature between the variables ; then on replacing the x's and y's by their
97] gordan's theorem 105
values in terms of the ^'s the first equation will be satisfied identically and the second equation will become a linear equation between the ^'s with integral coefficients.
Hence by the above reasoning every solution of the equation among the ^'s may be written in the typical form
^=T,7i+ 2^272 + ... +r,7<., where t = yi, ^ = 72, ••• t = ya^
typify the irreducible solutions, and the T's are all positive integers.
Now substitute these values for the t's in the expressions for the xs and y's and we find at once that
where the /c's and X,'s are fixed positive integers.
Thus the only possible irreducible solutions of the two equations are those typified by
X = K^, y ^= \,i\ X =^ K2, y ^^ A<2 J ... X ^ K(r , y ^ A,(, ,
for every other solution can be expressed as a linear combination of these.
If we had a third equation, on substituting for the xs and y's their values in terms of the T's the first two equations would be satisfied identically, and the third would become a linear equation among the T's. Then this equation in turn has only a finite number of irreducible solutions, and hence, reasoning exactly as before, we should find that the three equations given have only a finite number of irreducible solutions. The process can be manifestly extended to any number of equations, and hence our theorem is established. A formal proof by induction from (r — 1) equations to r equations could of course be easily given.
Ex. To find the irreducible solutions of the two equations
x-\-w=y + z)'
The irreducible solutions of the second equation are easily found since no letter can exceed 2 ; they are
x=\, y = l; x = \, 2 = 1; y = l, w=\; z=l, w = l;
variables not naentioned in a solution being zero.
106 THE ALGEBRA OF INVARIANTS [CH. VI
Hence the general solution of the second equation is a;=a + b, i/ = a + c, z=b+d, w = c+a, where a, 6, c, d are positive integers. The first equation now becomes
and for an irreducible solution a, 6, c cannot exceed 2. On trial we find the following irreducible solution
a = 2, c?=5; 6 = 2, d=\; c = l, d=\; a = l, 6 = 1, d=Z. Hence the general solution is
a=2a + S, 6 = 2/3 + S, C=y, c?=5a+/3 + y + 38. These values for a, 6, c, d give
a7 = 2a + 2i3 +2S\
y = 2a + y +
« = 5a + 3j3+ y + 4S 20 = 50+ /3 + 2y + 3S, as the general solution of the two equations.
The only possible irreducible solutions are accordingly ^=2, y = % z=5, w=5 x=2, y=0, 2 = 3, w=l x=0, y=l, 2=1, w = 2\ x=2, y = \, 2=4, xo=Z}
Of these the first is the sum of the third and fourth, while the foui-th is the sum of the second and third, so the only irreducible solutions are the second and third. In other words any solution of the two equations may be written in the form
x=2p, y = q, z = 3p + q, w=p + 2q,
where p, q are positive integers.
Ex. (i). Prove that the equation lx + 4y—3z has four irreducible solutions and that every solution of the two equations 7x + 4y = 3z, z+5w=2y can be written in the form
x=2a + c, y = 7a+l5b + Uc, 2 = 14a + 206 + 17c, w = 2b + c. Ex. (ii). Find the number of irreducible solutions of the equation
11 •" 2 2 ■" • • • "t" n n ^~ ^3C^
the a's being positive integers.
Ans. If all the a's are even there are n irreducible solutions, if r of
rir—V) the a's are odd there are « + — solutions. In an irreducible solution
at the most only two of the letters on the left are different from zero.
97-99] gordan's theorem 107
Ex. (iii). Prove directly that in an irreducible solution of the two equations
hyi+hy2+ ••• +Kyn=y+z
X is less than the greatest a, and y is less than the greatest h.
98. System of forms derived by transvection from two given systems. Consider two systems of binary' forms in the same variables a^i, x.^, viz.
J-i, A.2, ... Am, of orders a,, a^, ... a^ respectively
and Bi, B^, ... B^, of orders h^, b^, ... bn respectively;
we suppose each form written symbolically and denote by U, V two products of the types
A,-^A,-^ . . . Am'^'n, B/^Bl^ . . . B,fn
wherein all the exponents are either zero or positive integers.
The system C is said to be derived from the systems A and B by transvection when it includes all terms in all transvectants of the form
{U, V)y.
It is clear that some of the members of the system C are reducible, that is they can be expressed as rational integral functions of simpler meijibers of that system — in fact, if
then there are many terms in the transvectant
{U, V)y which are products of two terms, one belonging to the transvectant
and the other to the transvectant
(U,, V,)y^.
99. We can now enunciate and prove our first theorem, viz.
The number of transvectants of the form ( U, V)y which do not contain reducible terms is finite.
* It is of course assumed that y^, y^ are such that these transvectants are possible, e.g. y^ must not exceed the order of V-^.
108 THE ALGEBRA OF INVARIANTS [CH. VI
For suppose that any term of the transvectant
contains p symbols of the A's, not in combination with a symbol of the B's and a symbols of the B'& not in combination with a symbol of the J.'s, then we have
because each side of the first equation, for example, represents the total number of the symbols of the forms A which occur in the product U.
Now to each positive integral solution of the above equations in a, y8, p, o", 7 there correspond definite products U, V and a definite value of 7 and hence a unique transvectant. But as we have already remarked if the solution corresponding to (17, V)y be the sum of those corresponding to (U^, Fi)^' and {17.2, ^■2)'^^ then (U, V)y certainly contains reducible terms. Hence trans- vectants corresponding to reducible solutions always contain reducible terms and inasmuch as the number of irreducible solutions has been proved to be finite it follows that the number of transvectants not containing reducible terms is finite.
100. In actually finding the transvectants which do not contain reducible terms we may use the equations (I), but it is generally easier to proceed directly.
Suppose that the system A contains the single form /= a^^ and the system B the single form i = hi, then we have to consider transvectants
If 7 > 2/3 this transvectant vanishes, if 7 < 2y3 — 1 it contains terms of i (/", i^~^)t ; hence for an irreducible transvectant we must have 7 = 2/3—1 or 2/3, In the same way
7 :^ 5a and -^ oa — 4.
Again if a > 2, then for an irreducible transvectant 7 > 10 and hence /3 > 4, so that some terms may be reduced by means of (/^ i^Y. Thus we need only consider a = 0, a = 1 and a = 2.
For a = 0 we have i.
For a=l we have/ (/i), {f,i)\ {f,i-)\ {f,ij, {f,i^)\
99-101] gordan's theorem 109
For a = 2 we have {f\ i'Y, (/^ iy, {f\ i*y, (/^ i%
Now {f^, iy contains terms which are products of a term of (/, i'^ and (/, i)'^ and a like argument applies to
(/^ i% (A i% (A i'f>
so the only transvectants not containing reducible terms are
/ i if, i), if, if, (/ i% (/, i')\ (f, iJ. (A> iT- Ex. (i). If/ be any form of order 2n + l, then the transvectants
which do not contain reducible terms are 2?i + 4 in number.
Ex. (ii). Find the corresponding result when / is a form of even order. Ex. (iii). The only transvectants
where /[=aa;* and/2=6a;^, which do not contain reducible terms are
/i, /2, h (/i, i\ ifi, ^7, (/i> i'r> (/i' ^')s
(A, i), (/2> if, (A, i')\ if2% i'f-
101, Definition. The system of forms A is said to be complete when any expression derived by convolution from a product U of powers of the forms A is itself a rational integral function of the A's.
Thus for example the system of forms
H = (abfa^bx
t — {ahy {ca) hxC^
A = {aby(cdy{ac){bd)
is complete because any expression derived in the above manner is a covariant of/ and therefore a rational integral function of /, H, t and A. Again the system H and A included in the above is itself complete.
More generally the system A is said to be relatively complete for the modulus G consisting of a number of symbolical deter- minants when any expression derived by convolution from a product i7 is a rational integral function of the ^'s together with terms involving the modulus G.
110 THE ALGEBRA OF INVARIANTS [CH. VI
Thus the system consisting of a single form
is relatively complete for the modulus (aby, since any expression derived by convolution from a power of f can be transformed so that a factor (aby occurs in it.
Again for a quartic the system
/= a^^ H = (aby aA^ t = (aby (ca) aj)^^c^\
j = {bey {cay {aby
is relatively complete for the modulus {aby, for all covariants of / are rational integral functions of
f,H, t,j,i, where i = {aby.
We may extend our definition of relative completeness still further : a system A is said to be complete relatively for several moduli Gi, 0^ ... when any expression derived by convolution from a product C is a rational integral function of the A's together with terms involving one at least of the moduli G^, G^
It will be seen later (or it can be verified without difficulty) that in connection with any quantic a^^ = bx^ ... the single form
H = {aby a^^'-^b^''-''
is relatively complete with respect to the modulus {aby except when n = 4.
If n = 4 the complete system worked out for the form H shews that any expression derived by convolution firom a power of H is a rational integral function of H together with terms involving i or j.
That is H is relatively complete for the two moduli {aby and {bey {cay {aby.
It will be noticed that a complete system is relatively complete for any modulus or systems of moduli.
102. The system G derived by transvection from two given systems contains an infinite number of forms, but it is said to be a finite system when all its members can be expressed as rational integral functions of a certain finite number of them. More generally it is said to be relatively finite for a given modulus G
101-103] gordan's theorem 111
when every member of C can be expressed as a rational integi"al function of a certain finite number of them together with terms all of which involve the modulus G.
For example the number of covariants of a binary cubic is infinite but inasmuch as every one is a rational integral function of/, H, t and A the system of forms is said to be finite.
Again it will be seen later that every covariant of the binary w-ic
can be expressed in terms of/, H, t, where
t = {ahf (ca) a^-^h^'-^c^-"-
together with terms involving the factor {ahf. We should state this fact thus — The system of covariants of a binary n-\Q is relatively finite for the modulus {ahf.
103. Theorem. If the systems of forms A and B are both finite and complete, then the system derived from them by transvection is finite and complete.
(a) The system is finite.
In the proof of this theorem we shall consider the transvectants
{U, V)y in a certain order defined as follows : —
(i) Transvectants are taken in order of ascending total degree of the product UV in the coefficients of the forms involved in A and B.
(ii) Those for which the total degree is the same are taken in ascending order of indices.
Further than this the order is immaterial.
With this convention let T and T' be any two terms of the transvectant
{U, V)y, then {T-T') = L{U,T)y'
where 7' < 7 and U, V are derived by convolution from U, V respectively.
112 THE ALGEBRA OF INVARIANTS [CH. VI
But since the systems A and B are complete
U = F{A\
where F{A) is a rational integral function of the J.'s, that is, an aggregate of products of the type TJ, and ^ (jB) a similar function of the 5's.
Thus {iJ^yy
can be expressed as the sum of a number of transvectants in each of which the index is less than 7. By hypothesis all such transvectants have been examined before the one now under consideration and hence if all the (7's derived from previously considered transvectants can be expressed in terms of
Cj, Cj, ... Cy,
then all C's up to and including those derived from
can be expressed in terms of
Ci, C^a, ... Cry T,
where T is any term of the last transvectant. But if the transvectant
contain a reducible term, say T=T^T^, then inasmuch as T^, T^ must both arise from transvectants previously considered no term T need be added to the system
Cj, C2, ... Gf.
Thus in gradually building up a system of C's in terms of which all O's can be expressed we need only add a new member when we come to a transvectant containing no reducible term and then we need add only one new member. But the number of transvectants containing no reducible term is finite and hence a finite number of (7's can be chosen such that every other is a rational integral function of these, that is the system G is finite.
Remark. A set of C's in terms of which all others can be expressed rationally and integrally can be chosen in various ways, for any term may be selected from each transvectant containing no reducible terms. Further since the difference of two terms of a
103] gordan's theorem 113
transvectant can be expressed by means of terms of transvectants previously considered we may, instead of choosing a single term from any transvectant, take an aggregate of any number of such terms or even the transvectant itself, and it will still be true that every member of G can be expressed as a rational integral function of the members of our finite system.
(6) The finite system so constructed is complete.
i-iet Oj, L/2, ••' 0,.
be the finite system, then we have to prove that an expression W derived by convolution from any product of the form
is a rational integral function of C^, C^, ... C^.
Suppose that W contains p determinantal factors in which a symbol belonging to a form A occurs in combination with a symbol belonging to a form B.
Then W is a term in a transvectant
(u, vy,
where U contains only symbols of the A's and V only symbols of the B's, so that U is derived by convolution from a product U of the ^'s and V is derived by convolution from a product V of the B's.
Thus W={u, vy + tiuvy,
where p < p and UV are derived from U, V by convolution and therefore ultimately from U, V.
Now U=F(A),
V=^{B),
accordingly W can be expressed as an aggregate of transvectants of the form
(U, vy.
But we have just proved that every term of such a transvectant is a rational integral function of the C's and consequently W is also a rational integral function of them.
Hence the system is not only finite but complete.
G, & Y. 8
114 THE ALGEBRA OF INVARIANTS [CH. VI
104. Theorem. If a finite system of forms A, all the members of which are covariants of a binary form f include f and be relatively complete for the modulus H ; if further, a finite system B be relatively complete for the modulus G, and include one form Bi whose only determinantal factors are H, then the system C derived by transvection from A and B is relatively finite and complete for the modulus G.
As an example of the theorem let A consist of
f=cix= bx\ and B of the two forms
E = {aby aj)^, A = {abf (ac) {bd) {cd)\
Then A is relatively complete for the modulus (a6)^ § 88, and B is absolutely complete, being the complete system of the Hessian of the cubic ; hence according to the theorem the system derived by transvection should be absolutely complete. This is obviously true, for the new system contains /, H, t. A, where
t = {fH) = -{ohy{ac)b^c^\
and every possible member of the derived system is a covariant of /, therefore they are all rational integral functions of f H, t. A, which constitute the complete system of the cubic.
105. Lemma. If P be derived by convolution from a power off any term in the transvectant
(P, vy
can be expressed as an aggregate of transvectants of the type
{U, vy
in which the degree of U is at most equal to that of P.
(Throughout we shall use U, V as typical symbols for products of powers of the forms of A and B respectively.)
This statement is manifestly true when the degree of P is zero ; assuming it true when the degree of P is less than r we shall establish it when the degree is r.
In fact if T be a term in
(P, vy, T={P,vy+X{P, vy-
104-105] gordan's theorem 115
and since P, P are derived by convolution from a power of the form f which is contained in A,
P=^F{A) + HW*,
P = F'{A) + HW',
while V=^{B) + GZ=^{B), mod a
Hence T can be expressed as the sum of three parts ;
(i) transvectants of the type [F{A), ^{B)Y the degree of F{A) being r ;
(ii) transvectants of the type {Q, Vy, where Q is of the same degree as P and contains the factor H ;
(iii) terms containing the factor 0.
Now Q can be derived by convolution from
where s is less than r the degree of P ; therefore any term in
can be derived by convolution from
and is expressible in the form
S(P',5rr),
where P' is derived by convolution from /* and is of degree less than P. But by hypothesis every terra in these transvectants can be expressed as an aggregate
t{U, vy, modG^, for 57F=^(P), modG^,
where the degree of U is at most equal to s and therefore less than r.
On referring to the expression for T we see that T can be written in the form
^{U, vy, modG;
consequently the statement in the lemma can be completely established by induction.
* HW simply means a symbolical product containing the factor H.
8—2
116 THE ALGEBRA OF INVARIANTS [CH. VI
Cor. If the product P contain the factor H, then any term in
(P, Vy
can he expressed in the form
where the degree of U is less than that of P.
For P is now of the fonn Q just discussed, and any term in a transvectant
(Q, vy
can be expressed as a sum
t(U,vy
in which the degree of U is at most equal to s which is less than the degree of P.
106. The proof of the theorem is now the same in principle as that in § 103.
The transvectants are considered in the following order.
(i) In order of ascending degree of UV in the coefficients of/
(ii) Those for which the degree of UV is the same are taken in order of ascending degree of U.
(iii) Transvectants for which these two degrees are the same are taken in order of ascending index.
Further than this the order is immaterial. If T and T' be two terms in
{U, vy^ _
then T'-T=t{U, Vy\
where v <v.
But U=F{A) + liW,
therefore
T'-T=t[F{A), ^{B)Y + ^[HW, <^{B)Y, modG. Transvectants of the type
{FiA),^{B)y have been previously considered, for the degree of F(A) is the same as that of U and v <v; further by the lemma transvectants of the type
[HW^^iB)]"'
105-106] gordan's theorem ' 117
can be expressed in the form
where the degree of U' is less than that of HW, i.e. less than that of U. .
Thus T'-T can be written
2 ( U", Vy + t(U', Vy, mod G,
where the degree of U" is the same as that of U and v' < v, while the degree of U' is less than that of U.
Hence if all terms of transvectants considered previously to
{u, vy
can be expressed rationally and integrally in terms of
Gi, C2, ... Gp
(except for terms involving G); then all terms of transvectants up to and including
{U, vy
can be expressed in the form
F{C„G,,... Gp,n modG, where T is any term of the last transvectant. If the transvectant
(u, vy
contain a reducible term we may suppose it to be T, and since T—T^T^ where T^, T^ are terms of former transvectants, there is no need to add the term T to Cj, Cg, ... Cp.
It follows that in constructing a system of (7's in terms of which all (7s can be expressed we have to add a new member only when we come to a transvectant containing no reducible terms and then one only. The number of transvectants con- taining no irreducible terms is finite, § 99 ; hence if G^,G^, ... Gg be a series of terms one from each of this finite number of transvectants, any other member of the system G derived by transvection from A and B can be expressed as a rational integral function of G^G^, ... Gq together with terms involving the factor G; in other words, the system G is relatively finite for the modulus G.
Next the system
(7i, G^, ... Gq
is relatively complete with respect to the modulus G.
118 THE ALGEBRA OF INVARIANTS [CH. VI
For any term T derived by convolution from may be regarded as a term in a transvectant
(u, vy,
where U is derived by convolution from a product of the A's and V from a product of the B's.
Hence T can be expressed as an aggregate of transvectants
{U,vy-
while U=P can be derived by convolution from a power of / and
V=^{B), modG; therefore T^^\P,<^ {B)Y, mod G,
= X (P, vy, mod G, = t(U, vy, mod G. (Lemma.) Consequently, as has just been proved,
T = F{G„G„... Gq), modG, and the system is complete.
107. Cor. I. If the system B is absolutely complete, then the system derived by transvection from A and B is absolutely complete.
Cor. II. If the system B is complete for two moduli G and G' and contains a form whose only determinantal factors are H, then the derived system is complete for the two moduli G and G'.
To prove this we have only to write
B = F{B), modd {G,G')
instead of B = F(B), mod (G)
at every stage of the foregoing proof.
108. Gordan's Theorem. These long preliminary ex- planations are now at an end and the actual proof of the theorem does not present much difficulty.
Every co variant of a binary form
/=a^« = 6^" = etc.
is either a power of / or else contains a factor (aby, and hence the form f itself is a complete system with respect to the modulus (aby.
lOG-109] gordan's theorem 119
Assuming now that a system of co variants containing / and relatively complete for the modulus (ab)^ can be found we shall shew how to construct a system also containing f and relatively complete for the modulus (a6)^+^ The system relatively complete mod (aby'' is called -4^-1, and since every covariant can be derived from / by convolution it is a rational integral function of the forms in A^^i except for terms involving the factor {obY'.
To construct the system Aj^ when A^^^ is known we make use of the theorem of § 104.
We must therefore begin by constructing a system 5fc_i possess- ing the following properties :
(i) it contains the form {ahf^ a^^'-'^h^''-^,
(ii) it is relatively complete for the modulus (a6)^+^
Then the system derived from ^^-i and Bj^_i by transvection will be finite and complete with respect to the modulus (a6)^+^ and as it obviously contains f which is contained in ^^-i it is the system A^ required.
109. Accordingly we have now to shew how to construct the system Bj^^. .
There are three cases.
?i
I. If 2A?<- then any. form derived by convolution from a
power of H^. = {ab)'^ a^^'-'^hx'^-^'' is of grade {1k+l) at least and therefore of grade {2k + 2) since all symbols are now equi- valent.
Hence Hj^ is itself relatively complete for the modulus (a6)^+'^ and in this case th« system B^ consists of the single form
{ahf' a^-'^h^-'^. (§ 74.)
II. If 2A; > ^ then H^ = (ab)* a^^^^^h^'^-^ is of order less than
n, say m.
Now we suppose that the complete system of co variants for a form of order < w is known and we derive a system from Hk on the model of the complete system of a^"* as explained in i 79, 80.
120 THE ALGEBRA OF INVARIANTS [CH. VI
Neglecting terms containing (a6)'*+2 ^q gan replace each copied
form by a single term ; the system so derived is complete for the
modulus (a6)^+^ and is therefore the system Bj^^i required.
ft III. If 2k = ^ — a case which can only arise when w is a
multiple of 4 — we have a rather different state of things.
Here the form Hj^ = {ah)*ax^~^hx^~-^ is relatively complete for the two moduli
(a6)^+S {ahY' {hc)"^ {caf\
the latter being an invariant J, and hence by Cor. II. § 107 the system derived by trans vection from A]c_y^ and ^j._i is relatively complete for the moduli (a6)^+^ and J\ calling this system G^ for a moment we have
G, = F(C,) + J.P„ mod (abr^^ where Pi is a covariant of degree less than G^..
Further since Pj can be derived by convolution from / which is contained in G^, we have
P, = F, (Cfc) + /. P^, mod {ahf+\
where Pa is a covariant of degree less than Pj.
Proceeding in this way we see that Gk is a rational integral function of / and the forms in Gk together with terms involving the factor {ah)'*^^
Hence if we add J to the system G^ and call the total system Ajc it follows at once that A^ is relatively complete for the modulus
Therefore in every case, given the complete system mod (a6)^ we can construct that mod(a6)'*+^; but the system A^ is/, thence we find the system A^, then from that the system A^ and so on, in fact we can construct the system A^ relatively complete for the modulus (a6)^+='.
110. Consider now a little more closely what happens when we come to the end of the sequence of moduli {aby, {oh)*, {aby ..., and first let n be even and equal to Ig.
Then the system Ag^^ is relatively complete for the modulus {aby^, and the system Bg^i consists of the single invariant (ab)^ so that it is of course absolutely complete.
100-111] gordan's theorem 121
Hence the system derived from Ag^i and Bg_i by transvection is absolutely complete and it contains/, therefore it is the complete system of invariants and covariants ; further since Bg^i consists of a single invariant the complete system Ag consists of Ag_i and that invariant {aby^.
Secondly let n be odd and equal to 2g + l, then the system Ag_j^ can be constructed and it both contains /and is relatively complete for the modulus {ahy^.
The system J5^_i is derived from the quadratic
{aby^ a^bx
by the same convolutions as the complete system of the quadratic oi^ = ^^ is found from this form. This complete system being a^' and {ci^y the system Bg^^ consists of
{abyo aj}^, {abya {ac) (bd) (cdy^.
This system is relatively complete for the modulus {ahy^'^^ by § 109 II, and this being a vanishing invariant it follows that jB^_i is absolutely complete.
Hence the system derived from Ag_^ and Bg^^ contains / and is absolutely complete, that is it constitutes the complete system of/.
To recapitulate — the complete system mod {aby can be written down at once, then from that we deduce the complete system mod {ahy and proceeding step by step we can finally construct an absolutely complete system as the last step in our series.
We have therefore proved that the complete system is finite, for all the systems A^, A^, ... are finite, and we have shewn how to construct it on the assumption that the systems for forms of lower orders are known — the proof is thus inductive in its nature.
111. We shall illustrate the above process by applying it to the quadratic, cubic, and quartic.
(i) Quadratic. The system Aq is
and the system B^ is {aby, hence the complete system is
a^\ {aby.
122 THE ALGEBRA OF INVARIANTS [CH. VI
(ii) Cubic. Here Aq is
fz=.ai = hi = etc. and 5i is (abf a^h^, {aVf {ac) (bd) {cdf,
in fact H, {H,H)\
This system B^ is absolutely complete, therefore the system derived by transvection is the complete system.
It consists of
f,H,(H,Hy = A and {f'^,m)y.
Proceeding as in § 88 we can shew that the only irreducible transvectant is (/, H).
(iii) Quartic. Here A^ is
/=««' = &«;'.•., £o is H = (aby a^bi,
and this is complete modd (aby and (aby (bey (cay.
The system derived by transvection is
If 7 > 2 this has a term containing the factor (aby (acy which is congruent to zero modd (aby, (aby (bey (cay.
Hence we need take only 7 = 1 and thence only a = l, /3 = 1, and we find that
/ H. (f, H) is relatively complete modd (aby, (aby (bey (cay.
Therefore / H, (f, H), (aby (bey (cay
is complete mod (aby and is the system Ai.
Then Bi being the invariant i = (aby we have for the complete system
f,H,t = (f, H), i = (aby, j = (aby (bey (cay. 112. We shall now apply the principles of §§ 73, 76 to the
n
deduction of a complete system mod (aby for the binary form of order n.
The system ^o consists of
/= a,;" = fta;" = etc.
and £o of H = (aby a^-^ fca,"-^ ....
111-112] gordan's theorem 123
The system Ai is derived by transvection from Aq and Bq.
Now (/-, H^)y
has a term containing the factor {aVf{acf if 7 > 1, and since such
a term is
= 0 mod {ahy
the transvectant may be rejected.
If 7 = 1 the transvectant contains reducible terms unless a = yS = 1, and hence A-^ consists of
f,H,{f,H) = t The system Bi is
(ahy a^-^ h^-^
and -4.2 is derived by transvection from A^ and B^.
If the index of a transvectant be greater than two it contains a term having a factor {ahf {acf and this is
= 0mod(a6)«. (§70.)
We need only consider the cases in which the index is ^ 2, and since the order of each form in A^ is certainly greater than 2 (in
fact ^ ^ 4), products of forms may be rejected.
There remain transvectants of each form of Ai taken simply with
H^ = {ahYa^''-'h^'^.
For the future we shall only write down the determinantal factors of a covariant. »
Transvectants with / give rise to
{ahy {he), {ahy {hey. Those with H give
{ahy {he) {cdy, {ahy {hey {cdy,
and finally those with t give
{ahy {be) {edy {dc), {ahy {bey (cdy (dc).
Now by § 76
(ahy (bey (cdy = (ahy (cdy, mod (ahy ;
hence (ahy (hey (cdy (de)
being a term of {{aby (bey (edy, e^"}
124 THE ALGEBRA OF INVARIANTS [CH. VI
we have
(aby (bcf (cdy (de) = {(aby {cdy, e^% mod {aby,
for all expressions derived by convolution from
{aby {bey {cdy
are ' =0 mod {aby. (§ 73.)
Now a term of the last trans vectant is
{cdy . {aby {ae),
.-. {aby {bey {cdy {de) = {cdy . {aby {ae), mod {aby
and accordingly may be rejected.
Finally {aby {be) {cdy {de)
is reducible as being the Jacobian of a Jacobian, and the system A^
consists of
f, {aby, {aby {be),
{aby, {aby {be), {aby {bey, {aby (be) {cdy.
113. Before proceeding further we shall develope the results of § 76 by shewing that a symbolical product T containing the factor
(a6)^ {bey {cdy,
in which X is even and equal to fi + v, can in general be expressed in terms of covariants that are either reducible or of grade greater than X.
The above reduction of
^{aby{bey{cdy{de) is a case in point.
In fact r is a term of
{{aby {be)'' {cdy, 4>Y
which we write {T, jty.
Hence V = {T, j>y -^X{f,^y, p<p
= {T, 4>y + 2 {T, 4>y', mod (a6)^+\
since T derived by convolution from {ahY {bey {caf is of grade greater than \, § 73.
Again
T={aby.{cdy-\-C^^, (§76),
r = [{aby . {cdy, <f>]f> + {{aby . {cd)\ ^j"' + Cx+x.
112-114] GORDANS THEOREM 125
Now if 2n — 2X^p each of these transvectants contains terms having {cdy Cx"^'^ dx^~^ as a factor.
Hence
{{ahY.{cdY,^Y = (cdy {{ahf, <t>Y + 2 {{aby . (cdf, 0}', mod (abY+\ a < p,
and by continuation of this process we can express T entirely in terms of reducible covariants and covariants of grade greater than X; it suffices to remark that the index a diminishes at every step.
It is quite easy to see that the condition
2n-2\^p
is satisfied in all our cases — at any rate it will be in the course of the subsequent work.
114. Returning now to the general form, B2 consists of
Hs^iabyaaT-'ba;''-',
and A3 is derived by transvection from A2 and B2.
The argument used in evolving A^ and A.2 enables us to see
(i) that transvectants with index > 3 may be rejected,
(ii) thence that transvectants of products or powers of forms may be likewise rejected.
We are therefore left with transvectants of the forms of -4 2 taken simply with H^, the index being :}> 3.
Omitting Jacobians of Jacobians and forms having a factor
{aby (bey (cdy, where /j, + v^6, we have
from/ (aby (be), (aby (bey, (aby (bey;
H, (aby (be) (cdy, (aby (bey (cdy, (aby (bey (cdy;
(aby (be), (aby (bey (cdy (de), (aby (bey (cdy (de) \
(aby, (aby (be) (cdy;
(aby (be), none ;
(aby (bey, {aby (be) (cdy (dey ;
(aby (be) (cdy, none.
126 THE ALGEBRA OF INVARIANTS [CH. VI
Hence we have found for A3 the above ten forms in addition to those of A2. Putting aside the question as to whether any of these ten new forms are reducible, a continued repetition of the above process establishes the fact that all the forms of the system Aj^, relatively complete for the modulus
are included in the set
(abf {her (cdf {dey {eff" {fgY" ...»
where the exponents satisfy the followipg conditions :
(i) ^i^^h
(ii) \, \', \", . . . are all even,
(iii) \>X' + /M, V > \" + fjf, . . .,
(iv) no two of the exponents /*, fi', ... are equal to unity.
In fact
(ii) follows immediately from the way in which the covariants are formed.
(iii) results from the application of §§ 73, 76.
(iv) is the expression of the fact that the Jacobian of a Jacobian is reducible.
Ex. (i). If the orders to, n, p of the forms /, (f), yl^, be each greater than two, then
Ex. (ii). For a form whose order is greater than four the covariants {abf (be) {cd), {ab) (bcf (cd), {ahf {bcf {cd) all vanish identically.
Ex. (iii). If « > 5, then {bcf {caf {ahf aj"-* bx^-^c^"-*
= (ab)* {acf a«»-6 Jx""* Cx"-2 - \ {abf a^^-s bj'-^ . Cx\
Ex. (iv). If » > 4, then and if w > 5,
* Cf. Jordan, Liouville's Journal, 1876, 1879.
114] gordan's theorem 127
Ex. (v). For a form whose order is greater than three prove that
Hence replacing (/, /)* by i express (H, 3)^ as a linear combination of
Si, fii, /)',/'{/,/)' and finally express t^ in terms of the irreducible forms of the system.
Ex. (vi). Prove that all irreducible covariants of degree four and rank
not greater than - are included in
{abf^ (bcf {cdf
71
where 2X :^ - and X > /i > v.
Ex. (vii). In § 103 if no A be of order greater than m and no B be of order greater than ?i, then no form of the system C is of order greater than m + n — 2.
CHAPTER VII.
THE QUINTIC.
115. To obtain the complete irreducible system of covariants of the quintic, we follow step by step Gordan's proof of the finiteness. Let us briefly recapitulate.
The complete system of