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Bible Encyclopedias

1911 Encyclopedia Britannica

Algebraic Forms

The subject-matter of algebraic forms is to a large extent connected with the linear transformation of algebraical polynomials which involve two or more variables. The theories of determinants and of symmetric functions and of the algebra of differential operations have an important bearing upon this comparatively new branch of mathematics. They are the chief instruments of research, and have themselves much benefited by being so employed. When a homogeneous polynomial is transformed by general linear substitutions as hereafter explained, and is then expressed in the original form with new coefficients affecting the new variables, certain functions of the new coefficients and variables are numerical multiples of the same functions of the original coefficients and variables. The investigation of the properties of these functions, as well for a single form as for a simultaneous set of forms, and as well for one as for many series of variables, is included in the theory of invariants. As far back as 1 773 Joseph Louis Lagrange, and later Carl Friedrich Gauss, had met with simple cases of such functions, George Boole, in 1841 ( Camb. Math. Journ. iii. pp. 1-20), made important steps, but it was not till 1845 that Arthur Cayley (Coll. Math. Papers, i. pp. 8 0 -94, 95112) showed by his calculus of hyper-determinants that an infinite series of such functions might be obtained systematically. The subject was carried on over a long series of years by himself, J. J. Sylvester, G. Salmon, L. O. Hesse, S. H. Aronhold, C. Hermite, Francesco Brioschi, R.F.A. Clebsch, P. Gordon, &c. The year 1868 saw a considerable enlargement of the field of operations. This arose from the study by Felix Klein and Sophus Lie of a new theory of groups of substitutions; it was shown that there exists an invariant theory connected with every group of linear substitutions. The invariant theory then existing was classified by them as appertaining to " finite continuous groups." Other " Galois " groups were defined whose substitution coefficients have fixed numerical values, and are particularly associated with the theory of equations. Arithmetical groups, connected with the theory of quadratic forms and other branches of the theory of numbers, which are termed "discontinuous," and infinite groups connected with differential forms and equations, came into existence, and also particular linear and higher transformations connected with analysis and geometry. The effect of this was to co-ordinate many branches of mathematics and greatly to increase the number of workers. The subject of transformation in general has been treated by Sophus Lie in the classical work Theorie der Transformationsgruppen. The present article is merely concerned with algebraical linear transformation. Two methods of treatment have been carried on in parallel lines, the unsymbolic and the symbolic; both of these originated with Cayley, but he with Sylvester and the English school have in the main confined themselves to the former, whilst Aronhold, Clebsch, Gordan, and the continental schools have principally restricted themselves to the latter. The two methods have been conducted so as to be in constant touch, though the nature of the results obtained by the one differs much from those which flow naturally from the other. Each has been singularly successful in discovering new lines of advance and in encouraging the other to renewed efforts. P. Gordan first proved that for any system of forms there exists a finite number of covariants, in terms of which all others are expressible as rational and integral functions. This enabled David Hilbert to produce a very simple unsymbolic proof of the same theorem. So the theory of the forms appertaining to a binary form of unrestricted order was first worked out by Cayley and P. A. MacMahon by unsymbolic methods, and later G. E. Stroh, from a knowledge of the results, was able to verify and extend the results by the symbolic method. The partition method of treating symmetrical algebra is one which has been singularly successful in indicating new paths of advance in the theory of invariants; the important theorem of expressibility is, directly we exclude unity from the partitions, a theorem concerning the expressibility of covariants, and involves the theory of the reducible forms and of the syzygies. The theory brought forward has not yet found a place in any systematic treatise in any language, so that it has been judged proper to give a fairly complete account of it.' I. THE Theory Of Determinants.' Let there be given n 2 quantities all a,2 a13 ��� aln a21 a22 a23 ��� a2n a3, a32 a33 ��� a3n and an, an 3 ��� ann and form from them a product of n quantities ala a2 0 a37 ... anv, where the first suffixes are the natural numbers I, 2, 3, ...n taken in order, and a, 0, y, ... v is some permutation of these n numbers. This permutation by a transposition of two numbers, say a, 13, becomes 0, a, 7, ... v, and by successively transposing pairs of letters the permutation can be reduced to the form I, 2, 3, .. .n. Let k such transpositions be necessary; then the expression X(kal aa2 N a 3. Y ...a n v, the summation being for all permutations of the n numbers, is called the determinant of the n 2 quantities. The quantities a l a, a 2 Q ... are called the elements of the determinant; the term ( -) k alaa20a37...anv is called a member of the determinant, and there are evidently n! members corresponding to the n! permutations of the n numbers I, 2, 3, ... n. The determinant is usually written all a12 a13. �� aln a a2n and ant an3 ��� ann the square array being termed the matrix of the determinant.

A matrix has in many parts of mathematics a signification apart from its evaluation as a determinant. A theory of matrices has been constructed by Cayley in connexion particularly with the theory of linear transformation. The matrix consists of n rows and n columns. Each row as well as each column supplies one and only one element to each member of the determinant. Consideration of the definition of the determinant shows that the value is unaltered when the suffixes in each element are transposed.

1 Theorem

2 Theorem

3 Corollary

4 Minors of a Determinant

5 Observation

6 Corresponding Minors

7 Multiplication

8 Remark

9 Theorem

10 Determinants of Special Forms

11 Theorem

1 (1 +/-lD1+Fl2D2+�3D3+...) (X i X 2 X 3 ...) � Comparing coefficients of like powers of A we obtain DX1(X1X2X3...) = (X2X3...), while D 8 (X 1 X 3 X 3 ...) =o unless the partition (X3X3X3...) contains a part s. Further, if DA 1 DA 2 denote successive operations of DA 1 and DA2, DX1DA2(x1X2X2...)

2 1 + b l(1) + b (12) + b 2(2) +bi (13) + b 1b2(21) + b 3(3) +... +00 2 0 ..b qm (m qm m -1 qm-1 ...2 Q2 1 s1) -{-... 2 3 m = ealal+Q2a2.. +amam Expanding the right-hand side by the exponential theorem, and then expressing the symmetric functions of al, a2, ...a m, which arise, in terms of b1, b2, ...' b., we obtain by comparison with the middle series the symbolical representation of all symmetric functions in brackets () appertaining to the quantities p i, P2, P3,��� To obtain particular theorems the quantities a l, a 2, a 3 , ...a, n are auxiliaries which are at our entire disposal. Thus to obtain Stroh's theory of seminvariants put b1=0-1+a2+��.+0-m

2.1 Ex. gr. (Sa-3b)2(Sa

3 log (1 +aiox +aol)/+...+apgxPyq+.... From this formula we obtain by elementary algebra 1) ! p, g 5

3.1 The Polar Process

3.2 The Evectant Process



If the determinant is transformed so as to read by columns as it formerly did by rows its value is unchanged. The leading member of the determinant is alla22a33���ann, and corresponds to the principal diagonal of the matrix.

We write frequently 0 = alla22a33���ann = (ana22a33���ann)� If the first two columns of the determinant be transposed the ' The elementary theory is given in the article Determinant.

expression for the determinant becomes Z(-) k aitia2aa3y...anv, viz. a and 13 are transposed, and it is clear that the number of transpositions necessary to convert the permutation say...v of the second suffixes to the natural order is changed by unity. Hence the transposition of columns merely changes the sign of the determinant. Similarly it is shown that the transposition of any two columns or of any two rows merely changes the sign of the determinant.


Interchange of any two rows or of any two columns merely changes the sign of the determinant.


If any two rows or any two columns of a determinant be identical the value of the determinant is zero.

Minors of a Determinant

From the value of A we may separate those members which contain a particular element a ik as a factor, and write the portion aik A ik; A k, the cofactor of ar k , is called a minor of order n - i of the determinant.

Now a11A11= alla22a33...ann, wherein all is not to be changed, but the second suffixes in the product a 22 a 33 ...a nn assume all permutations, the number of transpositions necessary determining the sign to be affixed to the member.

Hence anAu = auk t a22a33...ann, where the cofactor of an is clearly the determinant obtained by erasing the first row and the first column.

a ll a33 ��� a32 a33 ��� a3n an2 an3 ��� ann Similarly A ik , the cofactor of aik, is shown to be the product of (-) i+k and the determinant obtained by erasing from A the ith row and k th column. No member of a determinant can involve more than one element from the first row. Hence we have the development A = a11A11 +a12Al2 +a13A13+��� +ainAin, proceeding according to the elements of the first row and the corresponding minors.

Similarly we have a development proceeding according to the elements contained in any row or in any column, viz.

A =ailAii+a12A12+a13A13+��� +ainAin) (A) � A = lk + a2kA2k +a3kA3k +��� +ankAnk This theory enables the evaluation of a determinant by successive reduction of the orders of the determinants involved.

Ex. gr. 2 0-5 3 l I - 5 3 I - 0 I 0 3 I +3 1 0 - 5 13 I-6 I-5j--3.21-51-3.1101 =3+30-30-0=3.

Since the determinant having two identical rows, and an3 an3 ��� ann vanishes identically; we have by development according to the elements of the first row a21Au+a22Al2 +a23A13+��� +a2nAin =0; and, in general, since a11A11+a12A12 +ai 3A13+�� � +ainAin = A, if we suppose the P h and k th rows identical a A +ak2 A 12 +ak3A13+��� +aknAin =0 ( k > i) .and proceeding by columns instead of rows, a li A lk +a21A2k + a 31A3k+���+aniAnk = 0 ( k .><.i) identical relations always satisfied by these minors.

If in the first relation of (A) we write ais = bis+cis+dis+��� we find that laisAis = IbisAis +Ec i sA i s +Zd is A is +... so that A breaks u p into a sum of determinants, and we also obtain a theorem for the addition of determinants which have rows in common. If we multiply the elements of the second row by an arbitrary magnitude X, and add to the corresponding elements of the first row, A becomes Zai,A18+XEa28A13 = Lia13A18 =A, showing that the value of the determinant is unchanged. In general we can prove in the same way the - Theorem. - The value of a determinant is unchanged if we add to the elements of any row or column the corresponding elements of the other rows or other columns respectively each multiplied by an arbitrary magnitude, such magnitude remaining constant in respect of the elements in a particular row or a particular column.


Every factor common to all the elements of a row or of a column is obviously a factor of the determinant, and may be taken outside the determinant brackets.

Ex. gr. a 2 y2 a2 /32_ a2 y2 a2 - ,9_a2 72 - a2 a 1 3 y = a fl - a y - a - y - a I 1111 0 0 (0 - a)(7 - a)I i ay 1 a I = (a - 7)(7 - a)I 130771-al = (a - a ) (7 a)(0-7). The minor Aik is aa, and is itself a determinant of order n-t. We may therefore differentiate again in regard to any element ars where r> i and the s e' " column of A is the s th or ( s = I) th column of Aik according as sZk. Hence, if Tri denote the number of transpositions necessary to bring the succession ri into ascending order of magnitude, the sign to be attached to the determinant arrived at by erasing the P h and r th rows and the k th and s th columns from A in order produce Aik will be - i raised to the power of Tri +Tks+i+k-Fr+s. Similarly proceeding to the minors of order n-3, we find that Aik a t k a A A is obtained from A by eras rs aaikaarsoatu to ing the Ph, rth, teh rows, the k th, s th, u th columns, and multiplying the resulting determinant by - i raised to the power T tri +Tusk +i+k+r+s- -t+u and the general law is clear.

Corresponding Minors

In obtaining the minor Aik in the form of a determinant we erased certain rows and columns, and we would have erased in an exactly similar manner had we been forming the determinant associated with A2:8, since the deleting lines intersect rk in two pairs of points. In the latter case the sign is determined by -I raised to the same power as before, with the exception that Tux., replaces Tusk; but if one of these numbers be even the other must be uneven; hence A ik = - Ais� rk Moreover aik a,, aikarsAik +aisarkAis Aik, rk aik ars rs where the determinant factor is giyen by the four points in which the deleting lines intersect. This determinant and that associated with Aik are termed corresponding determinants. Similarly p lines rs of deletion intersecting in p 2 points yield corresponding determinants of orders p and n-p respectively. Recalling the formula A =a11A11+a12Al2+a13A13+���+a1nA1n, it will be seen that a ik and Alk involve corresponding determinants. Since A lk is a determinant we similarly obtain Alk = a21Alk+� � � +a2,k-iAl,k +a2,k+lAl,k+ ���+a2 21 2,k-1 2, k +1 2,n and thence = Xalia2kAli where k; i,k 2k and as before A = a1, an A i> k i,k I ail, auk 12k an important expansion of A. Similarly ali a21 a31 A =E a ik a2k a3k A li i > k > r, z�k'r alr a2r air 23',!

and the general theorem is manifest, and yields a development in a sum of products of corresponding determinants. If the jth column be identical with the i ll ' the determinant A vanishes identically; hence if j be not equal to i, k, or r, a 11 a 21 a31 0 =I alk a2k a3k A11. alr a 2r a3r 31,!

Similarly, by putting one or more of the deleted rows or columns equal to rows or columns which are not deleted, we obtain, with Laplace, a number of identities between products of determinants of complementary orders.


From the theorem given above for the expansion of a determinant as a sum of products of pairs of corresponding determinants it will be plain that the product of A= (a ll, a22, ��� ann) and D = (b21, b 22, b nn ) may be written as a determinant of order 2n, viz.

a11 a21 a31. �� a n1 - 1 a12 a 22 a32 ��� an2 0 a13 a 23 a33 ��� an3 0 a3n �� � a nn 0 0 0 ... - 1000 ... 0 b11 b12 b13 ��� b1n 0 0 0 ... 0 b21 b 22 b23 .�� b2n 0 0 0 ... 0 b31 b32 b33 ��� b3n 0 0 0 ...0 bra b,, 2 b n3 .�. bnn Mult ply the i st, 2nd nth rows by b 11 , b 12, ... bin respectively, and Hence A11= 0 0 ... 0 -1 0 ... 0 0 -1 ... 0 _ Iabi - Cd for brevity.

a21 a22 all ��� a20 a21 a22 a23 ��� a20 a31 a32 a33 ��� a30 add to the (n+ I) th row; by b 21, b 22 ... b 2 n, and add to the (n+ 2)th row; by b31, b 32, ... ban and add to the ( n+3) rd row, &c. C then becomes a11b11+a12b12+���+ainbin, a21b11+a22b12+���+a2nbin, ��. anibll +an2b12+� �� +annbin a11b21+a12b22+��� +alnb2n, a21b21+a22b22+��� +a2nb2n, � � � ani b21 + a n2 b 22 + � � � +annb2n alib31+a12b32+���+ainb3n, a21b31+a22b32+���+a2nb3n, .�.a n lb 31 + a n2 b 32+ ��� +annb2n a ll b nl + a 12 b n2+ ��� + a ln b nn, a21bn1+a22bn2+�-�+a2nbnn, � � � ani b nl + a n2 b n2 +� � � +annbnn and all the elements of D become zero. Now by the expansion theorem the determinant becomes (-)1 +2+3+�.�+2nB.0 = (- I)n(2n +1) +nC =C.

We thus obtain for the product a determinant of order n. We may say that, in the resulting determinant, the element in the ith row and k th column is obtained by multiplying the elements in the kth row of the first determinant severally by the elements in the ith row of the second, and has the expression aklb11+ak2b12+ak3b13��� +aknbin, and we obtain other expressions by transforming either or both determinants so as to read by columns as they formerly did by rows.


In particular the square of a determinant is a deter minant of the same order (b 11 b 22 b 33 ...b nn) such that bik = b ki; it is for this reason termed symmetrical.

The Adjoint or Reciprocal Determinant arises from A = (a11a22a33 ...a nn ) by substituting for each element A ik the corresponding minor Aik so as to form D = (A 11 A 22 A 33 ��� A nn). If we form the product A.D by the theorem for the multiplication of determinants we find that the element in the i th row and k th column of the product is akiAtil+ak2A12 +��� +aknAin, the value of which is zero when k is different from i, whilst it has the value A when k=i. Hence the product determinant has the principal diagonal elements each equal to A and the remaining elements zero. Its value is therefore O n and we have the identity D.0 = A n or D It can now be proved that the first minor of the adjoint determinant, say B rs is equal to An-2a�.

From the equations a11xi+ a12x2+ a13x3 +��� = El, a21x1+a72x2+ a23x3+��� = 2, a3lxl+a32x2+a33x3+��� = 53, 0x1 =A111+A21E2+A31Er3+��� 0x2 = Al2E1 + A22E2+ A32Srr3+��� AX3 =A13E1+A23E2+A33E3+��� A n 1 E1 = B110x1 + B12Ax2+ B13Ax3+���, On - lt2 = B 21Ax1+ B220x2+ B230x3+��� An-15513 = B31Ax1 + B 32Ax2+B330x3+��� and comparison of the first and third systems yields B = An-2a rs = rs� In general it can be proved that any minor of order of the adjoint is equal to the complementary of the corresponding minor of the original multiplied by the h power of the original determinant.


The adjoint determinant is the (n - I) th power of the original determinant. The adjoint determinant will be seen subsequently to present itself in the theory of linear equations and in the theory of linear transformation.

Determinants of Special Forms

It was observed above that the square of a determinant when expressed as a determinant of the same order is such that its elements have the property expressed by aik = aki. Such determinants are called symmetrical. It is easy to see that the adjoint determinant is also 'symmetrical, viz. such that Aik=Aki, for the determinant got by suppressing the ith row and k th column differs only by an interchange of rows and columns from that got by suppressing the k th row and i th column. If any symmetrical determinant vanish and be bordered as shown below all a12 a13 Al a12 a22 a23 A2 a13 a23 a33 A3 Al A2 A3 � it is a perfect square when considered as a function of A 11 A2, A3. For since A 11 A 22 -Ar 2 =,,a 33, with similar relations, we have a number of relations similar to A 11 A 22 =AM 2, and either Ars = +11 (A rr A ss) or - (A r .A ss) for all different values of r and s. Now the determinant has the value - {AiA11+A2A22+A3A33+2A2A3A23+2A3AIA31+2A1A2Al2{ = -Eata r r-2EA r A 8 A rs in general, and hence by substitution {A I V A n+ A 211 A22+��� +A71 Ann}2.

A skew symmetric determinant has a,. =o and ars=-asr for all values of r and s. Such a determinant when of uneven degree vanishes, for if we multiply each row by - I we multiply the determinant by (- I ) n = -1, and the effect of this is otherwise merely to transpose the determinant so that it reads by rows as it formerly did by columns, an operation which we know leaves the determinant unaltered. Hence 0 = - O or ., =o. When a skew symmetric determinant is of even degree it is a perfect square. This theorem is due to Cayley, and reference may be made to Salmon's Higher Algebra, 4th ed. Art. 39. In the case of the determinant of order 4 the square root is Al2A34 - A 13 A 24 +A14A23.

A skew determinant is one which is skew symmetric in all respects,. except that the elements of the leading diagonal are not all zero. Such a determinant is of importance in the theory of orthogonal substitution. In the theory of surfaces we transform from one set of three rectangular axes to another by the substitutions ' X=' by+ cz, Y = a'x + b'y + c'z, Z =a"x+b"y-l-c"z, where X 2+Y2+Z2 = x2+ y2+z2. This relation implies six equations. between the coefficients, so that only three of them are independent. Further we find x=aX+a'Y+a"Z, y=bX z= cX+c'Y+ c"Z, and the problem is to express the nine coefficients in terms of three independent quantities.

In general in space of n dimensions we have n substitutions similar to X l = a11x1 +a12x2 + � � � + ainxn, and we have to express the n 2 coefficients in terms of Zn(n - I)i independent quantities; which must be possible, because X1+X2+..."IL Xn =xi+x2 +x3 +...+4.

Let there be 2n equations r }}?

= b11EE1 + b12 EE t 2 + b133 + ���, X2 = b21E1 + b 22E2 + b 23E3 + ���, X1 = b11E1+b21}5.2+b3,5tt3 +��� X2 = b12E1+b2'_S2+ b 3253 +��� where b rr = I and b rs = - b sr for all values of r and s. There are then 2n(n-I) quantities b rs . Let the determinant of the b's be Ab and B rs, the minor corresponding to b rs . We can eliminate the quantities S l, E2, ��� In and obtain n relations AbXi = (2B 11 - Ab)'�k1 +2B21x2+2B31x3+���, AbX2 = 2B12x1+ (2B22 - Ab) x2 +2B32x3+..., and from these another equivalent set Abx1 = (2B11 - X1 +2B12X2+2B13X3+���, Abx2 = 2B21X1+(2B22 - Ab)X2+2B23X3+���, and now writing 2Bii - Ab 2Bik - aii, Ob = aik, Ob we have a transformation which is orthogonal, because EX 2 = Ex2 and the elements aii, a ik are functions of the 2n(n- I ) independent quantities b. We may therefore form an orthogonal transformation in association with every skew determinant which has its leading diagonal elements unity, for the Zn(n-I) quantities b are clearly arbitrary.

For the second order we may take Ob - I - A, 1 1 +A2, and the adjoint determinant is the same; hence (1 +A2)x1 = (1-A 2)X 1 + 2AX2, (l +A 2)x 2 = - 2AX1 +(1 - A2)X2.

Similarly, for the order 3, we take 1 v Ab= -v 1 A =1 +x2 + 1 ,2 + � - A 1 and the adjoint is 1+A v +A� -� +Av -v +A� 1+11 2 A +/-tv pt+AvA +�v 1 +1,2 leading to the orthogonal substitution Abx1 = (1 +A 2 - / 22 - v 2) X l +2(v+A�)X2 +2(/1 +Av)X3 1bx2 = 2(A� - v)Xl+(1 +�2 - A2 - v2)X2 / +2(Fiv+A)X3 Abx3 = 2(Av +�)X1 +2(/lv-A)X2+(1+v2-A2- (12)X3.

Functional determinants were first investigated by Jacobi in a work De Determinantibus Functionalibus. Suppose n dependent variables yl, y2,���yn, each of which is a function of n independent variables x1, x2 i ���xn, so that y s = f s (x i, x 2 ,...x n). From the differential coefficients of the y's with regard to the x's we form the functional. determinant we derive and thence ' dx' n ay2 ?ay? - 2 ay2' ax, Ux2 ��� a y n ay. � Oxl d 2x 77n If we have new variables z such that zs=4s(yl, Y2,...yn ), we have also z s =1 Y 8(x1, x2,���xn), and we may consider the three determinants which i s 7xk, the partial differential coefficient of z i, with regard to k . Hence the produc J1 t theorem (21, Z2,...zn / (y1, Y2,...y.n) =  ? zl, z2,...zn) yl, y2,. ..yn xi, and as a particular case (y1, Y2,...yn) (x1, x2,...xn ) = 1.

x l, x 2,... x n / I l yl, y2,���yn Theorem.-If the functions y 1, y2,��� y n be not independent of one another the functional determinant vanishes, and conversely if the determinant vanishes, yl, Y2, ...y. are not independent functions of x1, x2,...xn.

Linear Equations.-It is of importance to study the application of the theory of determinants to the solution of a system of linear equations. Suppose given the n equations fl= = allxl +a12x2 + � � � + annxn = 0, f2 =a21x1+a22x2+���+a2nxn =0, fn =anlxl +an2x2+��� +annxn = 0.

Denote by A the determinant (a11a22���ann)� Multiplying the equations by the minors A l ,.., A2,,,,���Ani., respectively, and adding, we obtain x 1 (ai, Aig+a2p.A2lc+���+an�An�) =x�A=o, since from results already given the remaining coefficients of x 11' x 2 ,...x � 'i x�+I,...x, vanish identically.

Hence if A does not vanish x 1 = x 2 =... =x� = o is the only solution; but if A vanishes the equations can be satisfied by a system of values other than zeros. For in this case the n equations are not independent since identically Al�ft+ A2� f2+...+An�fn = 0, and assuming that the minors do not all vanish the satisfaction of ni of the equations implies the satisfaction of the nth.

Consider then the system of ni equations a21xi+a22x2+��� + a2nxn = 0 a31x1+a32x2+���+a3nx,, =0 an1x1 + an2x2 + � � � +annxn = 0, which becomes on writing xs = y 8, a21y1+ a 22y2 + � � � + a 2,n-lyn-1 + a 2n = 0 a3lyl +a32y2+��� +a3,n-lyn-i+a3n =0 an1 y1 +an2y2 +��� +an, n-lyn-1 +ann = 0.

We can solve these, assuming them independent, for the - i ratios yl, y2,...yn-i� Now a21A11 +a22Al2 � � � = 0 a31A11+a32Al2 +� �� +a3nAln = 0 an1Al1+an2Al2 +���+annAln =0, and therefore, by comparison with the given equations, x i = pA11, where p is an arbitrary factor which remains constant as i varies.

Hence_li y ` A 1n where A li and A li, are minors of the complete determinant (a11a22...ann)� an1 ant ���an,n-1 or, in words, y i is the quotient of the determinant obtained by erasing the i th column by that obtained by erasing the n th column, multiplied by (-r)i+n. For further information concerning the compatibility and independence of a system of linear equations, see Gordon, Vorlesungen fiber Invariantentheorie, Bd. i, � 8. Resultants.-When we are given k homogeneous equations in k variables or k non-homogeneous equations in k - i variables, the equations being independent, it is always possible to derive from them a single equation R = o, where in R the variables do not appear. R is a function of the coefficients which is called the " resultant " or " eliminant " of the k equations, and the process by which it is obtained is termed " elimination." We cannot combine the equations so as to eliminate the variables unless on the supposition that the equations are simultaneous, i.e. each of them satisfied by a common system of values; hence the equation R =o is derived on this supposition, and the vanishing of R expresses the condition that the equations can be satisfied by a common system of values assigned to the variables.

Consider two binary equations of orders m and n respectively expressed' in non-homogeneous form, viz.

f(x) = f =a o xm "- a l + a 2 xm-2 - ... = 0, 4(x) = 4) = box' -bix'-1+.b2xn-2-... = 0.

If al, a2, ...a, n be the roots of f=o, (1, R2, -Ai the roots of 0=o, the condition that some root of 0 =o may qq cause f to vanish is clearly R s, 5 =f (01)f (N2) � �;f (Nn) = 0; so that Rf,q5 is the resultant of f and and expressed as a function of the roots, it is of degree m in each root 13, and of degree n in each root a, and also a symmetric function alike of the roots a and of the roots 1 3; hence, expressed in terms of the coefficients, it is homogeneous and of degree n in the coefficients of f, and homogeneous and of degree m in the coefficients of 4.. Ex. gr. f = aox 2 -alx+a2 =0, �=box2 -blx+b2. We have to multiply a01; -alas+a2 by ao, -aif32+a2 and we obtain ao ( 3 - aoal(f31N2 +01133 ) +aoa2(SI+13) -i-a?31a2 - aIa2(31 + 02) + al, 131+02 = b, 131132 = b t'i +s2 = 2bob2, and clearing of fractions R 1,5 = (a o b 2 - a2 b o) 2 + (a i b o - aobi ) (aib2 - a2b1).

We may equally express the result as 4,(al)Y'(a2)...0 (am) =0, (a 8 -fa t) =0.

This expression of R shows that, as will afterwards appear, the resultant is a simultaneous invariant of the two forms.

The resultant being a product of mn root differences, is of degree mn in the roots, and hence is of weight mn in the coefficients of the forms; i.e. the sum of the suffixes in each term of the resultant is equal to mn.

Resultant Expressible as a Determinant.-From the theory of linear equations it can be gathered that the condition that p linear equations in p variables (homogeneous and independent) may be simultaneously satisfied is expressible as a determinant, viz. if a11x1 +a12x2 +... +al pxp = 0, a21x1 +a22x2 + � � � +a2pxp = 0, aplxl+ap2x2+...+appxp = 0, be the system the condition is, in determinant form, ( = 0; in fact the determinant is the resultant of the equations. Now, suppose f and 4 ) to have a common factor x--y, f(x) =f1(x)(x--y); 4,(x) =4,1(x)(x--y), f l and 41 being of degrees m-1 and ni respectively; we have the identity ch i (x)f(x) =fl(x)4,(x ) of degree m+n-I.

Assuming then 01 to have the coefficients B1, B2,...B,, and f l the coefficients A 1, A21...A,n, we may equate coefficients of like powers of x in the identity, and obtain m+n homogeneous linear equations satisfied by the m+n quantities B1, 2 ,...B n, A 1, A 2 ,...A m. Forming the resultant of these equations we evidently obtain the resultant of f and 4,. Thus to obtain the resultant of aox 3 +a i x 2 +a 2 x+a 3, 4, =box2+bix+b2 we assume the identity (Box+Bi)(aox 3 +aix 2 +a2x+a3) = (Aox 2 +Aix+ A 2) (box2+bix+b2), and derive the linear equations Boa ° - Ac b o = 0, Boa t +B i ao - A 0 b 1 - A 1 bo =0, Boa t +B 1 a 1 - A0b2 - A1b1-A2b° = 0, Boa3+Bla2 - A l b 2 -A 2 b 1 =0, B 1 a 3 - A 2 b 2 =0, = = (y l, y2,...ynl `x1, x2,...xnl for brevity.

yl, y 2,...yn ) (zl, z2,...zn z1, z 2, ���zn xi, 'X' 2,... x n/ yl, Y2,...y n j ' x 1, � Forming the product of the first two by the product theorem, we obtain for the element in the ith row and kth column aZ, ayl az i ayz azi ayn ayl + e +...+ where or as a21 a22 ���a2,i -1 a2,i +1 .��a2n a31 ���a3,i -1 a3,ti+ 1 ���a3n ���yi -)tin,and a7,2 ���a,,,i -1 an,i+1�' a21 a22 ���a2, -1 a32 -1 I. .... and by elimination we obtain the resultant ao 0 bo 0 0 al ao b1 bo 0 a 2 a i b 2 b 1 bo a numerical factor being disregarded.

a3 a 2 0 b 2 b1 0 a 3 0 0 b2 This is Euler's method. Sylvester's leads to the same expression, but in a simp er manner.

He forms n equations from f by separate multiplication by x ,...x, I, in succession, and similarly treats 4) with m multipliers I. From these m+n equations he eliminates the m+n powers xmE.-1, xm+n- 2,.. 1,' treating them as independent unknowns. Taking the same example as before the process leads to the system of equations acx 4 +alx 3 +a2x 2 +a3x =0, aox 3 +a1x 2 +a2x+a 3 = 0, box +bix -1-b2x =0, box' +b i x 2 -{-h 2 x = 0, box + b i x + b:: = 0, whence by elimination the resultant a 0 a 1 a 2 a 3 0 0 a 0 a 1 a 2 a3 bo b 1 b 2 0 0 0 bo b 1 b 2000 bo b 1 b2 which reads by columns as the former determinant reads by rows, and is therefore identical with the former. E. Bezout's method gives the resultant in the form of a determinant of order m or n, according as m is n. As modified by Cayley it takes a very simple form. He forms the equation .f()4(') -.f(x')4)(x) = o, which can be satisfied when f and 4 possess a common factor. He first divides by the factor x -x', reducing it to the degree m - I in both x and x' where m>n; he then forms m equations by equating to zero the coefficients of the various powers of x'; these equations involve the m powers xo, x, - of x, and regarding these as the unknowns of a system of linear equations the resultant is reached in the form of a determinant of order m. Ex. gr. Put (aox 3 -}-a l x 2 +a 2 x +a 3) ( box' +b1x'+b2) - (aox'3+aix'2+a2x'+a3) (box' + bix + b2) = 0; after division by x-x the three equations are formed aobcx 2 = aobix+aob2 =0, aobix 2 + (aob2+a1b1-a2bo) x +alb2 -a3bo = 0, aob2x 2 +(a02-a3bo)x+a2b2-a3b1 =0 and thence the resultant aobo ao aob2 aob 1 aob2+a1b1-a2bo alb2-a3b0 aob 2 a1b2 - a 3 bo a2b2 - a3b1 which is a symmetrical determinant.

Case of Three Variables.-In the next place we consider the resultants of three homogeneous polynomials in three variables. We can prove that if the three equations be satisfied by a system of values of the variable, the same system will also satisfy the Jacobian or functional determinant. For if u, v, w be the polynomials of orders m, n, p respectively, the Jacobian is (u 1 v 2 w3), and by Euler's theorem of homogeneous functions xu i +yu 2 +zu 3 = mu xv1 +yv2 +zv3 = /IV xw 1+y w 2+ zw 3 = pw; denoting now the reciprocal determinant by (U 1 V2 W3) we obtain Jx =muUi+nvVi+pwWi; Jy=�.., Jz=..., and it appears that the vanishing of u, v, and w implies the vanishing of J. Further, if m ' =' p, we obtain by differentiation 7+x =m (u;1-2xl. + v ?xl 1 + u l U1+v1V 1 + w1W1) � or T x0a,?_ ( m-I ) J+m(o .

aj Hence the system of values also causes to vanish in this case; dx and by symmetry aj and Fz also vanish.

CY The proof being of general application we may state that a system of values which causes the vanishing of k polynomials in k variables causes also the vanishing of the Jacobian, and in particular, when the forms are of the same degree, the vanishing also of the differential coefficients of the Jacobian in regard to each of the variables.

There is no difficulty in expressing the resultant by the method of symmetric functions. Taking two of the equations ax + +cz) x"' 1 +... =0, a'xn+ (b'y+c'z)xn-1+... = 0, we find that, eliminating x, the resultant is a homogeneous function of y and z of degree mn; equating this to zero and solving for the ratio of y to z we obtain mn solutions; if values of y and z, given by any solution, be substituted in each of the two equations, they will possess a common factor which gives a value of x which, corn bined with the chosen values of y and z, yields a system of values which satisfies both equations. Hence in all there are mn such systems. If, therefore, we have a third equation, and we substitute each system of values in it successively and form the product of the mn expressions thus formed, we obtain a function which vanishes if any one system of values, common to the first two equations, also satisfies the third. Hence this product is the required resultant of the three equations.

Now by the theory of symmetric functions, any symmetric functions of the mn values which satisfy the two equations, can be expressed in terms of the coefficient of those equations. Hence, finally, the resultant is expressed in terms of the coefficients of the three equations, and since it is at once seen to be of degree mn in the coefficient of the third equation, by symmetry it must be of degrees np and pm in the coefficients of the first and second equations respectively. Its weight will be mnp (see Salmon's Higher Algebra, 4th ed. � 77). The general theory of the resultant of k homogeneous equations in k variables presents no further difficulties when viewed in this manner.

The expression in form of a determinant presents in general considerable difficulties. If three equations, each of the second degree, in three variables be given, we have merely to eliminate the six products x , 2, z 2, yz, zx, xy from the six equations u = v = w = o = oy = = 0; if we apply the same process :to thesedz equations each of degree three, we obtain similarly a determinant of order 21, but thereafter the process fails. Cayley, however, has shown that, whatever be the degrees of the three equations, it is possible to represent the resultant as the quotient of two determinants (Salmon, l.c. p. 89).

Discriminants.-The discriminant of a homogeneous polynomial in k variables is the resultant of the k polynomials formed by differentiations in regard to each of the variables.

It is the resultant of k polynomials each of degree m-I, and thus contains the coefficients of each form to the degree (m-I)'-1; hence the total degrees in the coefficients of the k forms is, by addition, k (m - 1) k - 1; it may further be shown that the weight of each term of the resultant is constant and equal to m(m-I) - (Salmon, l.c. p. loo).

A binary form which has a square factor has its discriminant equal to zero. This can be seen at once because the factor in question being once repeated in both differentials, the resultant of the latter must vanish.

Similarly, if a form in k variables be expressible as a quadratic function of k -1, linear functions X1, X2, ... Xic-1, the coefficients being any polynomials, it is clear that the k differentials have, in common, the system of roots derived from X1= X 2 = ... = = o, and have in consequence a vanishing resultant. This implies the vanishing of the discriminant of the original form.

af Expression in Terms of Roots.-Since x+y y =mf, if we take cx any root x 3, y1, ofand substitute in mf we must obtain, y 1 C) zaZ1 �; hence the resultant of and f is, disregarding numerical factors, y,y2...y,,. 1 X discriminant of f = ao X disct. of f. Now f = (xy 1 - x i y) (xy 2 - x 2 y) ... (x y m - x m y), ar _ y1(x y 2 - and substituting in the latter any root of f and forming the product, we find the resultant of f and d, viz.

y 1 y 2 ... y m (xly2 - x2y1) 2 (x0,3 - x 3 yl) 2... (x rys - x8yr) 2...

and, dividing by y1y2...ym, the discriminant of f is seen to be equal to the product of the squares of all the differences of any two roots of the equation. The discriminant of the product of two forms is equal to the product of their discriminants multiplied by the square of their resultant. This follows at once from the fact that the discriminant is Mara s) 2 II(/3, -fis)2{II(ar-/3$)}2.

References For The Theory Of Determinants.-T.Muir's "List of Writings on Determinants," Quarterly Journal of Mathematics. vol. xviii. pp. 110-149, October 1881, is the most important bibliographical article on the subject in any language; it contains 589 entries, arranged in chronological order, the first date being 1693 and the last 1880. The bibliography has been continued, and published at various dates (vol. xxi. pp. 299-320; vol. xxxvi. pp. 171-267) in the same periodical. These lists contain 1740 entries. T. Muir, History of the Theory of Determinants (2nd ed., London, 1906). School treatises are those of Thomson, Mansion, Bartl, Mollame, in English, French, German and Italian respectively.-Advanced treatises are those of William Spottiswoode (1851), Francesco Brioschi (1854), Richard Baltzer (1857), George Salmon (1859), N. Trudi (1862), Giovanni Garbieri (1874), Siegmund Gunther (1875), Georges J. Dostor (1877), Baraniecki (the most extensive of all) (1879), R. F. Scott (2nd ed., 1904), T. Muir (1881).

II. THE Theory Of Symmetric Functions Consider n quantities a l, a 21 a 3 ,... an.

Every rational integral function of these quantities, which does not alter its value however the n suffixes I, 2, 3, ... n be permuted, is a rational integral symmetric function of the quantities. If we write (I +a i x) (I a 2 x) ... (I x n x) = I +a l x+ a 2 x 2 -{-... +ax n, al, a2, are called the elementary symmetric functions.

a 1 = al+a2+...+an =21a1 a2 = aia2+aia3+a2a3+... = Zaia2 1 -a1x +a2x 2 -a3x3+... 1 +hlx-+h2x2+h3x3-}-..., which remains true when the symbols a and h are interchanged, as is at once evident by writing -x for x. This proves, also, that in any formula connecting a 1, a 2, a 3 ,... with h 1 , h 2, h 3 ,... the symbols a and h may be interchanged.

Ex. gr. from h 2 = a i -a 2 we derive a 2 = h i - h2.

The function Zap 1 a 2 P2 n being as above denoted by a partition of the weight, viz. p 1 p 2 ...p n), it is necessary to bring under view other functions associated with the same series of numbers: such, for example, as P P3 P2 P4 Pn -2 /, /, /, F i a i 1 a 2 Fi a 1 a 2 ... a n - 2 - 3) (P2P4...pn_2)� The expression just written is in fact a partition of a partition, and to avoid confusion of language will be termed a separation of a partition. A partition is separated into separates so as to produce a separation of the partition by writing down a set of partitions, each separate partition in its own brackets, so that when all the parts of these partitions are reassembled in a single bracket the partition which is separated is reproduced. It is convenient to write the distinct partitions or separates in descending order as regards weight. If the successive weights of the separates w 1, w 2, w 3 ,... be enclosed in a bracket we obtain a partition of the weight w which appertains to the separated partition. This partition is termed the specification of the separation. The degree of the separation is the sum of the degrees of the component separates. A separation is the symbolic representation of a product of monomial symmetric functions. A partition, (pipipip2p2p3) = can be separated in the manner (p 1 p 2) (PIP2) (p1P3) = (1)12,2) 2 (plp3), and we may take the general form of a partition to be (pi i p2 2 p3 3 ...) and that of a separa tion (J 1) 1 1(J 2) 5 2(J 3) 1 3... when J 1, J2, J3... denote the distinct separates involved.

Theorem.- The function symbolized by (n), viz. the sum of the n th powers of the quantities, is expressible in terms of functions which are symbolized by separations of any partition (n"1n'2n'3...) 1 !

of the number n. The expression is (-) V1+V2+V3 +...(Y1+Y2+13+�..- 1)1 (n) n .Y2.Y3.... 3 + � � (71+,%2+_%3+...- 1)! U 2.1 U2) 1.2 0 3) /3..., j1!j2!j3!...

(J1) j i(J2) 72 (J3)? 3 ... being a separation of (n1 1 n' 2 n3 3 ...) and the summation being in regard to all such separations. For the particular case (W1n:2n:3...) = (1n) 1 2 3 (-) n n(n) = / ( To establish this write 1 +�Xi +1/ 2 X2+f2 3 X3+... = 11(1 +�alxl+� g al x2+�3a1x3+...), the product on the right involving a factor for each of the quantities a l, a2, a3..., and u being arbitrary.

Multiplying out the right-hand side and comparing coefficients X1 = (1)x1, X 2 = (2) x2+(12)x1, X3 = (3)x3+(21)x2x1+ (13)x1, X4 = (4) x 4+(31) x 3 x 1+(22) x 2+(212) x2x 1 +(14)x1, Pt P2 P3 P1 P2 P3 Xm=?i(m l m 2 m 3 ...)xmlxm2xm3..., the summation being for all partitions of m.

Auxiliary Theorem.-The coefficient of l l i xl2x13... in the product Xm1Xm2Xm3... is1 (J1) j1(-12)12 (J 3)23. - where(J i) j l (J2) 72 (J 3) j a separ A1. / L 2 ! �3!...

ation of (l Al l h2 1 A3 ... ) of specification (m"lmH2m"...), and the sum is for all such separations. 1 2 3 To establish this observe the result.

1 xv = (3)1'1 (21)"2 (13) 73 ,r i 77 � 2 n�2+3,r3 i and remark that (3)' r i (21)' r 2 (I 3)? r3 is a separation of (3'r1277'211r2+37r3) of specification (3Y). A similar remark may be made in respect of �1 1 �2 1 �3 ,3, 2 X,1, / 2 2 !Xm2' /23!X..

and therefore of the product of those expressions. Hence the theorem.

Now log (1+�X1 +/22X2+/�3X3 +���) =E log (1+/2aix1+22aix2-1-/23ax3+...) whence, expanding by the exponential and multinomial theorems, a comparison of the coefficients of �n gives (n) (-)v1+v2+v3+.. .-1 (PI +V2+V3+...- 1)! x1'1xv2xv3 n n n n Vd 1,2!1,3!.. 1 2 3..

_ - ) V1+v2+v3+...1 (111+112+Y3+... - 1) !Xv1Xv2Xv3 Y1!Y2!1,3!... n1 n 2 n 3 �� � and, by the auxiliary theorem, any term XmiXm2X, n3 ... on the right-hand side is such that the coefficient of x n ix n Zx n 3... in 1 "1142 P3 X ? X. is A1 4 4 ,!, 3 1 ... 1 ?�� 2 m3..

(J1)11(J2)12(J3)j3�.� jj!j2!j3!..� where since(m1 1 m2 2 m3 3 ...) is the specification of (J1)j1(J2)j2(J3)j3..., � l +�2+/23+��� =ii +j2+j3+���� Comparison of the coefficients of x:14243... therefore yields the result (-) V1+v2+v3+... (P i +Y2+t' +...-1)! () n VI!Y2!P3!...

) j1+j2+j3+..� (J1+ j2 +j3+...-1)!/T1)?1(J2)72 (J 3)/3..., j11j2!j3!... ?.1 for the expression of Za n in terms of products of symmetric functions symbolized by separations of ( n 1 1n 2 2n 3 3) Let (n) a, (n) x, (n) X denote the sums of the n th powers of quantities whose elementary symmetric functions are a l, a 2, a31���; x 1, x2, x31..; X1, X2, X3,... respectively: then the result arrived at above from the logarithmic expansion may be written (n)a(n x) = (n)x, exhibiting (n) $ as an invariant of the transformation given by the expressions of X1, X2, X3... in terms of x 1, x2, x3,�� The inverse question is the expression of any monomial symmetric function by means of the power functions (r) = sr. Theorem of Reciprocity.-If �1 P2 "3 01 Q 2 7 3 Al A 2 A3 X m1 X m2 X m3 ... = ...+O(s i s 2 s 3 ...)xl1x12x13...+..., where 0 is a numerical coefficient, then also O ?2 0.3 P1 P2 P3 Al A2 A3 +.

X,1X82>$3...=...+8(m m m ...)x 11 x 12 x13......

1 2 3 We have found above that the coefficient of (x 1 1 x 12 x 13...) i n the product XmiXm2X m3 ... is �1!�2!�3!

'1 +� �.(11+j2+j3+... -1)!

(1)/1(12) 2(13)73....

(J1)ji(J2)72(J3)13��� jl!j2!j3!...

'?^ the sum being for all separations of 1 A1112 ` / a3 3 ...) which have the specification (m41 m2 2 m3 3 ...). We can multiply out this expression so as to obtain a series of monomials of the form 9(sl is2 2 s3 3 ...). It can be shown that the number 0 enumerates distributions of a certain nature defined by the partitions (,�i,�2...), (sT1s°2...), 1212 an = a 1 a 2 a 3 ... an.

The general monomial symmetric function is a P1 a P2 a P3. aPn 1 2 3 ' the summation being for all permutations of the indices which result in different terms. The function is written (plp2p3�4n) for brevity, and repetitions of numbers in the bracket are indicated by exponents, so that (p1p1p2) is written (p1p 2). The weight of the function is the sum of the numbers in the bracket, and the degree the highest of those numbers.

Ex. gr. The elementary functions are denoted by (1), (12), (13), ... (1n), are all of the first degree, and are of weights I, 2, 3,...n respectively. Remark.-In this notation (0) = Eai = (i n); ( 02) _ za l a2 = (2);... (0B) = (e), &c. The binomial coefficients appear, in fact, as symmetric functions, and this is frequently of importance.

The order of the numbers in the bracket (p l p 2 ...p n) is immaterial; we may therefore always place them, as is most convenient, in descending order of magnitude; the numbers then constitute an ordered partition of the weight w, and the leading number denotes the degree.

The sum of the monomial functions of a given weight is called the homogeneous-product-sum or complete symmetric function of that weight; it is denoted by h.; it is connected with the elementary functions by the formula 1 7r1l7r2!7r3! x3 (ll'1T2...) and it is seen intuitively that the number 0 remains unaltered when the first two of these partitions are interchanged (see Combinatorial Analysis). Hence the theorem is established.

Putting x1= I and x 2 = x 3 = x 4 = ... = o, we find a particular law of reciprocity given by Cayley and Betti, (1m1) t(1(1 n1 2) �2 (1.3)�3... = ... +ti ( Si 1S2 ?S3 3 ...) -f -..., (PO v1(1s2)a2(1.3)v3... _ ...+o(mi and another by putting x i = x 2 = x3= ...' =I, for then X. becomes hm, and we have h,�,,ih,�,,2hm3... _ ... +tir (S? 1 S 2 2 S 3 3 ...) +..., ?1 ?2 ?3 _ � l �2 �3 h S h S2 h 83 ... -. +o (m l m2 m3) +..., Theorem of Expressibility. - " If a symmetric function be symboilized by (A�v...) and (X1X2X3..�), (�i/-12�3���), (v1v2v3...)... be any partitions of X, respectively, the function isexpressible by means of functions symbolized by separation of X1A2X 3. � � / 1111-2113. � � P1 v2 v3...) � For, writing as before, Xm 'Xm 2 Xm '= zzo(SQls:2s73...) xi'x12x13..., 1 2 3" 1231 2 3 = EPxi l x A2 x A3, P is a linear function of separations of(/ 1 / 2 A2 / 4 3 3 ...) of specification (m"`1m�2m"`3...), and if X; 1 X 3 2X8 3 ' .. = ?P'xilx12xi 3 P' is a linear 1231 2312 ???

function of separations of (li'12 2 13 3 ...) of specification (si 1 s 22 s 33) Suppose the separations of (11 1 13 2 1 3 3 ... ) to involve k different specifications and form the k identities �1s � s Al A 2 A3 .. Xm1sXm2sXm3s... = EP x tl x t2 x t3 ... (S - 1 , 2, ...k), where (m�lsm"`2sm"`38...) is one of the k specifications. The law of reciprocity shows that p(s ) = zti (m 1te2tmtL3t) t=1 st It 2t 3t viz.: a linear function of symmetric functions symbolized by the k specifications; and that ( ) St =ti ts. A table may be formed expressing the k expressions Pa l), P(2),...P(1) as linear functions of the k expressions ( m"`'sm�2sm�3s...), s =1, 2, ...k, and the numbers BSc occurring therein is 2s 3s possess row and column symmetry. By solving k linear equations we similarly express the latter functions as linear functions of the former, and this table will also be symmetrical.


" The symmetric function (m �8 m' 2s m �3s ...) whose is 2s 3s partition is a specification of a separation of the function symbolized by ( li'l2 2 l3 3 ...) is expressible as a linear function of symmetric functions symbolized by separations of (li 1 12 2 13 3 ...) and a symmetrical table may be thus formed." It is now to be remarked that the partition (/,A.1/2)1/42/A38...)can be derived from (m"13m�2sm"`38...) 1 2 3 is 2s 3s by substituting for the numbers mi., m 231 m 331 ... certain partitions of those numbers ( vide the definition of the specification of a separation). Hence the theorem of expressibility enunciated above. A new statement of the law of reciprocity can be arrived at as follows: Since.

P(s) _ /ll8!/12s!/23s!...

t =1 (J1)Jl (J2)?2(J /3... ots(mlllsmtA2sm�3s...), j1 !j2 j3... ls 2s 3s where tist =tit8. Theorem of Symmetry. - If we form the separation function (J2) j1!j2!13!...

appertaining to the function (li'l32l3... ), each separation having a specification m" ` ' 8 m �2s m �38 multiply b P (is 2s 3s .��), P Y by ls! /t2s! / 38 !... and take therein the coefficient of the function (mi t tm7t t m 31 t ...), we obtain the same result as if we formed the separation function in regard to the specification (m� It t'tm2 32tm"`l3t...), multiplied by Alt!! /let! /1 3 1!... and took �1a � therein the coefficient of the function (mis m� 2s Es m 3s 3s ...)., take(li 1 l2 2. ..)=(214); (m ?88m288...) = (321); (m ?i t m2L t...)=(313); we find (21)(12)(1)+(13)(2)(1) =...+13(313)+..., (21) (1)3=...+13(321)+...

The Differential Operators. - Starting with the relation (1 + a i x) (1 +a 2 x)... (1 +a n x) = 1 +a 1 x+a 2 x 2 +... +a�xn multiply each side by I +px, thus introducing a new quantity A; we obtain (1 +a1x ) (1+a2x)...(1 -Fanx)(1+,ux) = 1+(a1 +1a)x + (a2+1aa1)x2+... so that f (al, a 3, a3,.�.an) =f, a rational integral function of the elementary functions, is converted into f(a1 +12, a2+ p a1,... a n +I la n -i) = f+/ldlf +?`id2f ` `3 d3f+...  ?. 1 1 where laan and di denotes, not s successive operations of d1, but the operator of order s obtained by raising d l to the s th power symbolically as in Taylor's theorem in the Differential Calculus.

Write also s l d1= D, so that f(a i a2+ p al, ) =f +FLDif +F4 2 D2f + t i 3 D 3 f -}-....

The introduction of the quantity p converts the symmetric function 1 2 3 into (XiX2X3+...) -Hu Al (X 2 A 3 .-) +/l02(X1X3.�.) +/103(A1X2.�.) +....

Hence, if f(ai, a 2, ...a n) _ (?i?2%?3���), 1 2 3 +,01(X2A3...) +02(X1X3.�.) +IlA'(XlX2.�.) +...

(1 +/-lD1+Fl2D2+�3D3+...) (X i X 2 X 3 ...) � Comparing coefficients of like powers of A we obtain DX1(X1X2X3...) = (X2X3...), while D 8 (X 1 X 3 X 3 ...) =o unless the partition (X3X3X3...) contains a part s. Further, if DA 1 DA 2 denote successive operations of DA 1 and DA2, DX1DA2(x1X2X2...)

(%3...), and the operations are evidently commutative.

Also D n D n 2 D;3 (,,{{,,11*1,/,?*2,/,Tr3) = I, and the law of o eration of the p2 X13 ... ['2 3 ... p operators D upon a monomial symmetric function is clear. We have obtained the equivalent operations 1 +/lDi+ p2 D2+/ 13D 3 - F ... = exp,udl where exp denotes (by the rule over exp ) that the multiplication of operators is symbolic as in Taylor's theorem. di denotes, in fact, an operator of order s, but we may transform the right-hand side so that we are only concerned with the successive performance of linear operations. For this purpose write as = a08+ aiaas+i+a2aas+2+....

It has been shown (vide " Memoir on Symmetric Functions of the Roots of Systems of Equations," Phil. Trans. 1890, p. 490) that exp(mldl +m2d2+m3d3+...) = exp (Midi +M2d2+M3d3+...), where now the multiplications on the dexter denote successive operations, provided that pp t exp(MiE+M2 2+M3E3+...) +mlH+m2V+m3S3+..., being an undetermined algebraic quantity.

Hence we derive the particular cases 1 1 expel ' =exp(d1 -2d2+5d3 - ...); exp/ld 1 = exp(Ad1p2d2 +/13d3 - ...), and we can express D. in terms of dl, d 2, d 3 ,..., products denoting successive operations, by the same law which expresses the ele mentary function a s in terms of the sums of powers s l, s 2, s3,...

Further, we can express d 8 in terms of Dl, D 2, D3, ... by the same law which expresses the power function s, in terms of the elementary functions a 1, a2, a3,...

Operation of 'D.' a Product of Symmetric Functions. - Suppose f to be a product of symmetric functions f i f 2 ...f m . If in the identity f =f l f 2 we introduce a new root A we change a 8 into a8+/la8_l, and we obtain (1 +AD1 2 D2+... +AsDs ...) p Di p2 D2+... -} p3D8 ...) fl X (1 +/lDl+�2D2+...+Asps+...) f2 X.

X (1 +PD1+12D2+...+�8D8+...) fm, and now expanding and equating coefficients of like powers of /t D 1 f - Z(Difi)f2f3. , D2f =I(D2f1)f2f3�, D 3 f =F(D3f1)f2f3... f m +Z(D2f1) (Dif2) f 2 f fm, the summation in a term covering every distribution of the operators of the type presenting itself in the term.

/ll8!/l2s!/138... (J1)11(J2)12(J3)/3... jl !j2!j3!...

ost m" m

lt 2t 3t

Writing these results Dif = D(1)f, D = D(2)f+D(l2)f, D3f = D(3)f+ (21)f+ D(13)f, s =1 (J1)11(J3)12(J3)13... j1!j2!j3!... where (J1) 11 (J2) 12 13. .. is a separation of (11 1 12213 3 ...) of specification ( mM'8m"`28m"`3s... ), placing s under the summation sign to denote the is Zs 3s specification involved, 141t412t!p31!...

1 a a a a d =aal+a laa2 a2aa3+... +an we may write in general D s f = ZD(p l p 2 p 3 ��) the summation being for every partition (piP2p3...) of s, and D(p iP2 p 3 ...)f being =2 (Dpifi)(DP2f2) ( DL'h3f3)f4...f,n. Ex. gr. To operate with D2 upon (213) (214) (15), we have D (2)f = (13) (214) (15) + (213) (14) (15), D c1 2)f = (122) (213) (15) +(213) (213) (14) + (212) (214) (14), and hence D2f = (214) (15) (13) +(213) (15) (14) +(213) (212) (15) +(213)2(14) +(214) (212) (14).

Application to Symmetric Function Multiplication.-An example will explain this. Suppose we wish to find the coefficient of (52413) in the product (20(2' 4)(0). (15).

Write (213) (214) (15) =... +A(524) (13) +...; then D5D1D1 (213) (214) (15) =A; every other term disappearing by the fundamental property of D8. Since we have: D2D?(1 4)(1 4)(13) =A Dg34 (13)+2(14)(13)(12)} =A D 2 D3 12(1)()+7(13)(1)+2(14)()+6(13)(12)} =A D712(1)3=A.

where ultimately disappearing terms have been struck out. Finally A=6.12=72.

The operator d1= aoaai+aiaa2+a20a3+... which is satisfied by every symmetric fraction whose partition contains no unit (called by Cayley non-unitary symmetric functions ), is of particular importance in algebraic theories. This arises from the circumstance that the general operator Ao,a0aa1 + ialaa2 + 2a2 a 3 +...

is transformed into the operator d 1 by the substitution (ac, al, a2, ���as, ���) _ (ao, Xoai, X 6 X i a 2, ���, XcX1..%s_las,���), so that the theory of the general operator is coincident with that of the particular operator d1. For example, the theory of invariants may be regarded as depending upon the consideration of the symmetric functions of the differences of the roots of the equation aox n - (i) a i x n - 1 + (z) a 2 x n 2 - ... = 0; and such functions satisfy the differential equation aoaa i +2a0a 2 +3a 2 aa 3 +... +na n _ i aa n = 0. For such functions remain unaltered when each root receives the same infinitesimal increment h; but writing x-h for x causes ao, a1, a 2 a3,... to become respectively ao, ai+hao, a2+2ha1, a 3 +3ha 2, ... and f(ae i a5, a 2, a3,...) becomes f+h(aoaai +2alaa2+3a2aa3+...) f, and hence the functions satisfy the differential equation. The important result is that the theory of invariants is from a certain point of view coincident with the theory of non-unitary symmetric functions. On the one hand we may state that non-unitary sym metric functions of the roots of aox n - a l x n - 1 -{-a 2 x n - 2 - ... =o, are symmetric functions of differences of the roots of aox n - 1!(n)a4xn-1+2!()a2xn-2-... = 0; and on the other hand that symmetric functions of the differences of the roots of aox n (7)alxn-1+ (z)a2xn-2-... =0, are non-unitary symmetric functions of the roots of a xn-a l xn 1 a2 x n-2 -... = 0.

0 1! +2!

An important notion in the theory of linear operators in general is that of MacMahon's multilinear operator (" Theory of a Multilinear partial Differential Operator with Applications to the Theories of Invariants and Reciprocants," Proc. Lond. Math. Soc. t. xviii. (1886), pp. 61

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Bibliography Information
Chisholm, Hugh, General Editor. Entry for 'Algebraic Forms'. 1911 Encyclopedia Britanica. 1910.

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