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.

Multiplication

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.

Remark

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.

Theorem

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, (alla22...app) = 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 ...an�' 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, ...an 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 ...an 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 3...is 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.

Theorem

" 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 ...).

Ex.gr., 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, ...an+Ilan-1) =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...)

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 ...fm 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. ..fm , D2f =I(D2f1)f2f3�..fm+2(Difi)(D1f2)f3...fm, D 3 f =F(D3f1)f2f3... f m +Z(D2f1) (Dif2)f3...fm+Z(D3f1) f 2 f fm, the summation in a term covering every distribution of the operators of the type presenting itself in the term.

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. 61-88). It is definied as having four elements, and is written the coefficient of a0 o a1 a2 2 ... being !

mi, ! . The operators ko.ki.k2 aoaai+alaa2+��., a00a i +2a11, 2 +��� are seen to be (I, o; 1, I) and (I, I; I, I) respectively. Also the operator of the Theory of Pure Reciprocants (see Sylvester Lectures on the New Theory of Reciprocants, Oxford, 1888) is (4, 1;2,1) =2 4a 0 ea 1 +10acaiaa 2 +6(2aoaz+a 2 1) 0 9a3+... � It will be noticed that (�, v; m, n) =p(1, 0; m, n)+v(0, 1; m, n).

The importance of the operator consists in the fact that taking any two operators of the system (I l, v; m, n); (Ill, v l : m l, n1), the operator equivalent to (I l , v; m, n ) (111, v 1; ml, n1) - (i l l , v1; ml, n1) (/l, v; m, n), known as the " alternant " of the two operators, is also an operator of the same system. We have the theorem (I I, v; m, n) (/l l, v l; ml , n i) - (Il l, P 1; m l, n ') (/l, v; m, n) = (11, vl; ml, ni); where 1 /l1= (ml +m-1) ml (/l +nlv) - u-2 Cu '+nvl) 1 1 m-1 1 m1-1 vl =(n -n)vv-E ml / lY- m /lv, m i =7111+m-I, n1=nl+n, and we conclude that qua " alternation" the operators of the system form a " group." It is thus possible to study simultaneously all the theories which depend upon operations of the group. Symbolic Representation of Symmetric Functions.-Denote the s 8 s elementar symmetric function a s by al a 2 a3 ...at pleasure; then, Y y si, ,si,... p, taking n equal to 00, we may write 1 +aix +a2x2 +... _ (1 + p ix) (1 + P2x) ... = a l z = e a2z =e.3.=...

where s s a i a 2 a3 = =..

Further, let 1 -1-b i x+ b 2 x 2' +... +bmx m = (1 +Q 1 x) (1 +0 2 x)... (1 +umx); so that 1 +alal+a2a1 +... = (1 +Plat) (1 +P2(71)... = ePlal, 1 + a i Q 2+ a 2 0 2 +... _ (1 +PiQ2) (1 +P2(72)... =e2a2, 1 +aiam.+a2am+... = (1 +Plain) (1 = er,nam; and, by multiplication, II (1 +ala+a2a2+...) = II (1-}-biP+b2P 2 +... +bmP"`), a = e?l a' 1 °2 a 2 +.. +om a m .

Denote by brackets () and [ ] symmetric functions of the quantities p and a respectively. Then 1111 + a i[ 1 ]+ a i [ 12 1+a2[ 2 ]+ a 7 [ 13 ] +ala2[ 21 ]+a3[3]+-� + a p1 a p2 a P 3 �� .ap rn[Y1 p 2t' 3 ... i'mJ +-� .

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

Ex. gr. 14(22) with m =2 must be a term in eQial+?2a2= eri (a1-a2>=...-[-a1(a1-a2)4+... and since b2 =at we must have (22) =24(al-a2)4 = 24(a i+ a 2) -6(a? a2+ ala2)+4a2a2 =2a4-2ala3+a2 as is well known.

Again, if a i, a 2, a 3 ...a m , be the t " roots of -1, b 1 = b 2 =... = b n_1 =o and b.= I, leading to 1 + (m) + (m 2) + (m 3) +... = ea lal+a2az+. .+omam (m8) =ms!(alai+a2a2+... +amaa.)sm, (ll, v; m P -"O a an + (l l + v) (ll +2 v) (m (11 +3v) +...], m - 2 2  !2 a 0 a 1 aan +2 ! 1 ! 1 ! a o -2 a1a2 ! 3 !a7-3ap ? aan +2 (m-1) ! 1 ! a0 a3 + (m-2) (m -m1)! 11 ! ao -ialaan+l m ! m _i m ! -1)!1 ! ao (m -2) m !

m! +(m -3) D 5(213) (214) (15) - (13) (14) (14), as= and and we see further that (alai +a2a2+...+amam) k vanishes identically unless (mod m). If m be infinite and 1 + b i x + b 2 x 2 +... (1 + a i x) (1+ = s i z we have the symbolic identity +02712+0.3x3+... ePl g l + P2P2 + P31 3 3 -f -.. �, and (alai +(72a2+a3a3+�� �) P = (Pith +P2t2 +P3f 3 3+ � � �) P � Instead of the above symbols we may use equivalent differential operators. Thus let =a10a0+2a20al+3a30a2+...

and let a, b, c, ... be equivalent quantities. Any function of differences of S a, S b, S c ,... being formed, the expansion being carried out, an operand ao or bo or co ... being taken and b, c,... being subsequently put equal to a, a non-unitary symmetric function will be produced.

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

Sc) = (Sa-23aSb +3b) (Oa - Se) =Sz - 23QSb+303 b - SQS c +23a3 b 3c - StSc = 6a 3 - 4a2b1 +2a,b 2 - 2a2c1 +2alblci - 2b2c1 =2 (al - 3a1a2+3a3) = 2 (3) .

The whole theory of these forms is consequently contained implicitly in the operation S. Symmetric Functions Several Systems Quantities. - It will suffice to consider two systems of quantities as the corresponding theory for three or more systems is obtainable by an obvious enlargement of the nomenclature and notation.

Taking the systems of quantities to be / al, a2, a3,...

132, 03,�.� we start with the fundamental relation (1+alx+aly)(1+a2x +t2Y) (1+a3x +03y)... = 1 +alox +aoly +a20x 2 +auxy +aG2y2 +... P y q +... As shown by L. Schlafli 1 this equation may be directly formed and exhibited as the resultant of two given equations, and an arbitrary linear non-homogeneous equation in two variables. The right-hand side may be also written /? /? /?

1+Eaix+Esiy+ /al a2x 2 +Malt2xy -Z01023,2+��� The most general symmetric function to be considered is E 41 041 8424-3033..� .conveniently written in the symbolic form (pigi p2g2 p3go...)� Observe that the summation is in regard to the expressions obtained by permuting then suffixes I, 2, 3, ...n. The weight of the function is bipartite and consists of the two numbers Ep and Eq; the symbolic expression of the symmetric function is a partition into biparts (multiparts) of the bipartite (multipartite) number Ep, Eq. Each part of the partition is a bipartite number, and in representing the partition it is convenient to indicate repetitions of parts by power symbols. In this notation the fundamental relation is written (l + a i x +01Y) (I + a 2x+l32Y) (1 + a3x+133y)... = 1 +(l A x +(01) y +(102) x2 +(1001)xy+(512)3,2 +(103)x 3 +(10201)x i y+(10 O12)xy2+ (013)y3+... where in general a pg = (10 P 010).

All symmetric functions are expressible in terms of the quantities ap g in a rational integral form; from this property they are termed elementary functions; further they are said to be single-unitary since each part of the partition denoting ap q involves but a single unit.

The number of partitions of a biweight pq into exactly i biparts is given (after Euler) by the coefficient of a ,z xPy Q in the expansion of the generating function 1 - ax. 1 - ay. 1 - axe. 1 - 1aye. 1ax3.1ax2y. 1 - axy2.1 - ay3...

The partitions with one bipart correspond to the sums of powers in the single system or unipartite theory; they are readily expressed in terms of the elementary functions. For write (pq) =s� and take logarithms of both sides of the fundamental relation; we obtain slox +soot' = + (3ly) S20x 2 +2siixy+s02y 2 = E(aix+(3 ly) 2, &C., and siox+SOly - (S 20 x2 + 2s ii x y+ s ooy 2) +...

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

7r corresponding to Thomas Waring's formula for the single system. The analogous formula appertaining to n systems of quantities which Vienna Transactions, t. iv. 1852.

expresses s pg ,... in terms of elementary functions can be at once written down.

We can verify the relations s 30 -a310 -3a 20 a 10 + a30, S 21 - 02100 01 -a 2C a 01 -0 11 0 10 021 The formula actually gives the expression of q) by means of separations of (10P01'), which is one of the partitions of (pq). This is the true standpoint from which the theorem should be regarded. It is but a particular case of a general theory of expressibility.

To invert the formula we may write 1 +aiox+aoly+... +apgxPyq+... = exp {(siox+Solt') - s20 x 2+ 2siixy+S02y2)+���}, and thence derive the formula ? /,) (-)P+4-laP4 (p i+ g l - 1) ! C '" 1 S (p2 +q21)t ? 7r 2 (-)?,rl ,rl 7r2pl lql ')C)C p2 !g2 ! � �� 7r1! 72 !...s7114h sP242...

which expresses the elementary functions in terms of the single bipart functions. The similar theorem for n systems of quantities can be at once written down.

It will be � shown later that every rational integral symmetric function is similarly expressible.

The Function hpg. - As the definition of h pg we take 1 + nlox+naly+... +n,gxPyq+...

1 -(1aix - Rly) (1-a2x-R2y)...' and now expanding the (P1 right-hand side _ I ql)(P 2 +1721..Q1 /2172�..), h pg - pi p2 / J L' the summation being for all partitions of the biweight. Further writing 1 +hlox+holy+...+ hpgx P y {-...

1a i ox +... + (-) P+q a pg x P y +..., we find that the effect of changing the signs of both x and y is merely to interchange the symbols a and h; hence in any relation connecting the quantities pg with the quantities a pg we are at liberty to interchange the symbols a and h. By the exponential and multinomial theorems we obtain the results) 1,r -1 (E7r) ! Aal Ar2 7R1! 7 R 2L.�� P141 P242..� And In This A And H Are Interchangeable.

(pi+qi - 1 )! ('1 (p2+172-1)! 1,r 2 S�2 pi! qi! S ] l p2! q2!.�� S ...7f1! 7r2!...SPIQYP242..� Dif f erential Operations. - If, in the identity 1 (1 +anx = 1+aiox+aoly+a20x 2 +allxy+a02y 2 +..., we multiply each side by (I -�-P.x+vy), the right-hand side becomes 1 +(aio+1.1 ') x +(a ol+ v) y +...+(a p4+/ 1a P-1,4+ va Pr4-1) xPyq - - ...; hence any rational integral function of the coefficients an, say f (al ° , aol, ...) =f exp(�dlo+vdol)f d a P-1,4, dot = dapg The rule over exp will serve to denote that i udio+ vdo h is to be raised to the various powers symbolically as in Taylor's theorem. Writing D = gi d od p! 1 exp(Adlo + vdol) = (1+/oD10+ v Doi +..�+ VQ +.�.)f; now, since the introduction of the new quantities 1.1., v results in the addition to the function (plglp2g2p3g3...) of the new terms A PI Pg1 (p 2q2 p 3g3���) +/ AP2Pg2 (p 1 g 1P343 ...)+/ Z3vg3 (p l g i p 2 g 2 ...)+ �, we find DP141(plqip2q2p3q3���) = (p 2 q 2 p 3 q 3���), and thence D P141 D P242 D P343 ��. (p g p ,g p ,g3 ���) = I; while D rs f =o unless the part rs is involved in f. We may then state that D pg is an operation which obliterates one part pq when such part is present, but in the contrary case causes the function to) 171-1-(E7r-1)!7r1 a?2 an! 7r 2 ! ... P141 P242 ��� -1 hp, - hpg = is converted into where dlo = d a P,q-1 - dapg vanish. From the above D p4 is an operator of order pq, but it is convenient for some purposes to obtain its expression in the form of a number of terms, each of which denotes pq successive linear operations: to accomplish this write d ars and note the general result exp (mlodlo+moldol +... +mp4dp4 +...) =exp Mp g dp 4+�� .

where the multiplications on the leftand right-hand sides of the equation are symbolic and unsymbolic respectively, provided that m P4, M P4 are quantities which satisfy the relation exp (M14+Moir+...+Mp4EpnP+...) =1+mic -Fmoif+...+mp,eng+...; where E, n are undetermined algebraic quantities. In the present particular case putting m 10 = 1 2, mot= v and m P4 =o otherwise M10t+M01n+...+Mpot P n 4 +... =log (1 +�t+vn) M P4 = (_)p+4 -1(p+g 1)!�p p 4; p!g! and the result is thus exp(Mdlo+vdol) = {�die+vdol- 2 (� 2 d2 +2�vd11+ v2d02)+...{ =1 +,D10+vD01+... +1.0v4Dp4+...; and thence p010+ v d01 - (� 2d 20+ 2 � vd 11 +v 2 d02) +��� = log (1+IuD10+PDc.1+...+�pv4Dp4+...).

(-) Dp4= P+4-1 ,w (p11 -1)! " 1 ? (p2 +g2- 1) ! +1,q p l !gll p2!g2!

?

)?n -1 d" �" lrl!7r2!... d d the last written relation having, in regard to each term on th right-hand side, to do with 17r successive linear operations. Recalling the formulae above which connect s P4 and a m , we see that dP4 and Dp q are in co-relation with these quantities respectively, and may be said to be operations which correspond to the partitions (pq), (10 P 01 4) respectively. We might conjecture from this observation that every partition is in correspondence with some operation; this is found to be the case, and it has been shown (loc. cit. p. 493) that the operation 1 1 d P? 41 d p1 42 ... (multiplication symbolic) ?r1! ?2,�..

corresponds to the partition (p1g1' rl p2g2 n2 ...). The partitions being taken as denoting symmetric functions we have complete correspondence between the algebras of quantity and operation, and from any algebraic formula we can at once write down an operation formula. This fact is of extreme importance in the theory of algebraic forms, and is easily representable whatever be the number of the systems of quantities.

We may remark the particular result (-) p + p q! d p4sp4 +Dp4(pg)+1; d P4 causes every other signle part function to vanish, and must cause any monomial function to vanish which does not comprise ,one of the partitions of the biweight pq amongst its parts.

Since dp4+(-)P+T1(p +q qi 1)!dd4, the solutions of the partial differential equation d P4 =o are the single bipart forms, omitting s P4 , and we have seen that the solutions of p4 = o are those monomial functions in which the part pq is absent.

One more relation is easily obtained, viz.

=d P 4 lodp+1,4 -holdp,4+1+...+(-)r+shrsdp+r,4+s+.. daP4 References For Symmetric Functions.-Albert Girard, In- -vention nouvelle en l'algebre (Amsterdam, 1629); Thomas Waring, Meditationes Algebraicae (London, 1782); Lagrange, de l'acad. de Berlin (1768); Meyer-Hirsch, Sammlung von Aufgaben aus der Theorie der algebraischen Gleichungen (Berlin, 1809); Serret, Cours d'algebre superieure, t. iii. (Paris, 1885); Unferdinger, Sitzungsber. d. Acad. d. Wissensch. i. Wien, Bd. lx. (Vienna, 1869); L. Schlafli, " Ueber die Resultante eines Systemes mehrerer algebraischen �leichungen," Vienna Transactions, t. iv. 1852; MacMahon, " Memoirs on a New Theory of Symmetric Functions," American 1 Phil. Trans., 1890, p. 490.

Journal of Mathematics, Baltimore, Md. 1888-1890; " Memoir on Symmetric Functions of Roots of Systems of Equations," Phil. Trans. 1890.

III. THE Theory Of Binary Forms A binary form of order n is a homogeneous polynomial of the nth degree in two variables. It may be written in the form n n-1 2 ax 1 +bx1 x2 +cx 1 x 2 + ...; or in the form n n n=1 n n-2 2 +(1)bx x2+ ?

1112 which Cayley denotes by (a, b, c, ...)(xi, x2)n (i),(2)��� being a notation for the successive binomial coefficients n, 2n (n-I),.... Other forms are n-1 n-2 2 ax +nbx x +n(n-i)cx x +..., 1121 2 the binomial coefficients C) being replaced by s!(e), and n 1, n-1 1 n-2 2 ax 1 +l i ox l 'x 2 + L ?cx 1 'x2+..., the special convenience of which will appear later. For present purposes the form will be written a0x 1 +(7)a1x1=1 x2+ C 2)o'2x12 x 2 +...+anx2, the notation adopted by German writers; the literal coefficients have a rule placed over them to distinguish them from umbral coefficients which are introduced almost at once. The coefficients a 01 a1, a2,..�an, n+I in number are arbitrary. If the form, sometimes termed a quantic, be equated to zero the n+I coefficients are equivalent to but n, since one can be made unity by division and the equation is to be regarded as one for the determination of the ratio of the variables.

If the variables of the quantic f(x i , x 2) be subjected to the linear transformation x1 = a12Et2, x2 = a21E1+a2252, E1, being new variables replacing x1, x 2 and the coefficients an, all, a 21, a22, termed the coefficients of substitution (or of transformation), being constants, we arrive at a transformed quantic f% 1tn n n-1 n-2 52) = a S +(1)a11 E 2 + (2)a2E1 E 2 +��� in the new variables which is of the same order as the original quantic; the new coefficients a, a, a'...a are linear functions 0 1 2 n of the original coefficients, and also linear functions of products, of the coefficients of substitution, of the nth degree.

By solving the equations of transformation we obtain rE1 = a22x1 - a12x1, r = - a21x1 + allx2, aua12 where r = I = anon-anon; a21 a22 r is termed the determinant of substitution or modulus of transformation; we assure x 1 , x 2 to be independents, so that r must differ from zero.

In the theory of forms we seek functions of the coefficients and variables of the original quantic which, save as to a power of the modulus of transformation, are equal to the like functions of the coefficients and variables of the transformed quantic. We may have such a function which does not involve the variables, viz.

F(a ' a ' a ,...a) =r A F(ao, a1, a2,���an), 0 1 2 n the function F(ao, al, a2,...an) is then said to be an invariant of the quantic gud linear transformation. If, however, F involve as well the variables, viz.

F(-1-1 -1 t a a 0, a l, a 2 ,... ;51, 2) = r F(ao, al, a2,...; xi, x2), the function F(a 01 a 1, a 2 ,... x i, x 2) is said to be a covariant of the quantic. The expression " invariantive forms " includes both invariants and covariants, and frequently also other analogous forms which will be met with. Occasionally the word " invariants " includes covariants; when this is so it will be implied by the text. Invariantive forms will be found to be homogeneous functions alike of the coefficients and of the variables. Instead of a single quantic we may have several f(ao, a1, a2...; x1, x2), 4 (b o, b1, b2,...; x1, x2), ... which have different coefficients, the same variables, and are of the same or different degrees in the variables; we may transform them all by the same substitution, so that they become _, _, _, _, _, _, f(a °, a, a 2 ,...; (b 0, b, b 2 ,...; 1, S2),.... If then we find F ( a, a 1, a 2,...b 0, b, b 2 ,...,. .. �; S = r A F(a 0, 711, a2,���bo, b l, b 2,���9���; xl, x2), viz.

)- ( p+4-1 (p - - q -1)!dpq+ ?l -) 1)!D'1 DT2 p!g! ....

From these formulae we derive two important relations, dp4 = or the function F, on the right which multiplies r, is said to be a simultaneous invariant or covariant of the system of quantics. This notion is fundamental in the present theory because we will find that one of the most valuable artifices for finding invariants of a single quantic is first to find simultaneous invariants of several different quantics, and subsequently to make all the quantics identical. Moreover, instead of having one pair of variables x i, x2 we may have several pairs yl, y2; z i, z2;... in addition, and transform each pair to a new pair by substitutions, having the same coefficients a ll, a12, a 21, a 22 and arrive at functions of the original coefficients and variables (of one or more quantics) which possess the abovedefinied invariant property. A particular quantic of the system may be of the same or different degrees in the pairs of variables which it involves, and these degrees may vary from quantic to quantic of the system. Such quantics have been termed by Cayley multipartite.

Symbolic Form.-Restricting consideration, for the present, to binary forms in a single pair of variables, we must introduce the symbolic form of Aronhold, Clebsch and Gordan; they write the form Iln n n-1 n-1 n n n aixi+a2x2) = 44+(1) a l a 2 x 1 x2+...+a2.x2=az wherein al, a2 are umbrae, such that n-1 n-1 n a 1, a 1 a 2 ,...a 1 a 2 , a2 are symbolical respreentations of the real coefficients �o, ai,... an_1 i a n, and in general a n-k a 2 is the symbol for Q k. If we restrict ourselves to this set of symbols we can uniquely pass from a product of real coefficients to the symbolic representations of such product, but we cannot, uniquely, from the symbols recover the real form, This is clear because we can write n-1 n-2 2 2n-3 3 a1a2 =a l a 2, a 1 a 2 = a 1 a2 while the same product of umbrae arises from n n-3 3 2n-3 3 aoa 3 = a l .a a 2 = a a 2 .

1 1 Hence it becomes necessary to have more than one set of umbrae, so that we may have more than one symbolical representation of the same real coefficients. We consider the quantic to have any n number of equivalent representations a- b n -c n So that a 1 -k a 2 = b l -k b 2 - c 1 -k c 2 = ... = a k; and if we wish to denote, by umbrae, a product of coefficients of degree s we employ s sets of umbrae.

n-1 2 Ex. gr. We write;L 22 = a 1 a 2 .b 1 n-2 b2s 3 n - 3 3 n-3 3 n-3 3 a 3 = a 1 a 2 .b 1 b 2 .c 1 c2, and so on whenever we require to represent a product of real coefficients symbolically; we then have a one-to-one correspondence between the products of real coefficients and their symbolic forms. If we have a function of degree s in the coefficients, we may select any s sets of umbrae for use, and having made a selection we may when only one quantic is under consideration at any time permute the sets of umbrae in any manner without altering the real significance of the symbolism. Ex. gr. To express the function aoa2 - _ which is the discriminant of the binary quadratic aoxi -+-2a1x2x2-+a2x2 = ai =1, 1, in a symbolic form we have 2(aoa 2 -ai) =aoa2 +aGa2 -2 a1 � al = a;b4 -}-alb? -2ala2blb2 = (aib2-a2b1)2.

Such an expression as a l b 2 -a 2 b i, which is aa 2 ab 2 aa x 2 2 ax1' is usually written (ab) for brevity; in the same notation the determinant, whose rows are a l, a 2, a3; b2, b 2, b 3; c 1, c 2, c 3 respectively, is written (abc) and so on. It should be noticed that the real function denoted by (ab) 2 is not the square of a real function denoted by (ab). For a single quantic of the first order (ab) is the symbol of a function of the coefficients which vanishes identically; thus (ab) =a1b2-a2bl= aw l -a1ao=0 and, indeed, from a remark made above we see that (ab) remains unchanged by interchange of a and b; but (ab), = -(ba), and these two facts necessitate (ab) = o.

To find the effect of linear transformation on the symbolic form of quantic we will disuse the coefficients a 111 a 12, a21, a22, and employ A1, I�1, A2, �2. For the substitution rr xl =A 11 +1 2 12, 52=A21+�2E2, of modulus A1 �i = (Al�.2-A2�1) = (AM), A 2 �2 the quadratic form a k xi -1-2a 1 x i x 2 +a 2 4 = x =f (x), becomes A41 +2A1E16 =At = OW, where Ao = aoA i +2a1AiA2 +a2Az, _ _ A 1 = ao A l�l +ai(A1/.22+A2�1) +7,2X2/22, A2 = ao�l +2a1�1/�2 +a 2�2 � We pass to the symbolic forms a:= (aixi+a2x2) 2, A 2 = (A 151+ A 26) 2/ by writing for ao, al, a2 the symbols ai, a 1 a 2, a? A 1, A2 � Ai, A 1 A 2, A2 and then Ao = al Ai+2a1a2AIA2+a2 A2 - (a1A1+a2A2) 2 = a?, A l = (a 1 A 1 +a2A2) (al�l +a2�2) = aAa�, A 2 = (al�l +a2/-12) 2 = aM; so that A = aa l +2a A a u 152+aM5 2 = (aA6+a,e2)2; whence A1, A 2 become a A, a m, respectively and ?(S) = (a21+a,E2) 2; The practical result of the transformation is to change the umbrae a l , a 2 into the umbrae a s = a1A1 +a2A2, a � = a1/�1 + a21=2 respectively.

By similarly transforming the binary n ic form ay we find Ao = (aI A 1 +a2 A2) n = aAn A l = (alAi - I -a 2 A 2) n1 (a1�1 +a2m2) = aa a � - A i n-1 A2, n-k k n-k k n-k k A = (al l+a2A2) (al�1+a2�2) = a A � =A 1 A2, so that the umbrae A1, A 2 are a A, a � respectively. Theorem. -When the binary form a y = (alxl +a2x2)n is transformed to A;. = (A11+A22)n by the substitutions 51 = A l, E1+�1 2, 52 = A2E1+�2E2, the umbrae Al, A2 are expressed in terms of the umbrae al, a 2 by the formulae A l = Alai +A2a2, A2 = �la1 +�2a2� We gather that A1, A2 are transformed to a l, a 2 in such wise that the determinant of transformation reads by rows as the original determinant reads by columns, and that the modulus of the transformation is, as before, (A / .c). For this reason the umbrae A1, A 2 are said to be contragredient to xi, x 2. If we solve the equations connecting the original and transformed unbrae we find (A �) (- a 2) =A i( - A 2) + �'1A1, (A �) a1 = A2(- A2)+�2A1, and we find that, except for the factor (A /), -a 2 and +ai are trans formed to -A 2 and +A i by the same substitutions as x i and x 2 are transformed to i and E2. For this reason the umbrae -a 2, a l are said to be cogredient to 5 1 and x 2. We frequently meet with cogredient and contragedient quantities, and we have in general the following definitions:-(i) " If two equally numerous sets of quantities x, y, z,... x', y', z',... are such that whenever one set x, y, z,... is expressed in terms of new quantities X, Y, Z, ... the second set x', y', z', ... is expressed in terms of other new quantities X', Y', Z', .... by the same scheme of linear substitution the two sets are said to be cogredient quantities." (2) " Two sets of quantities x, y, z, ...; E, n, i, ... are said to be contragredient when the linear substitutions for the first set are x =A1X+u1Y-}-v1Z-?--..., y = A2X+,u2Y +v2Z �..., Z = A 3 X +�3Y -1v 3 Z - -..., and these are associated with the following formulae appertaining to the second set, .`"?. = A1?+A277+A3? +..., H =/.G1rr+�27]+�3? + ���, Z = v16+v2%/+v3" +���, wherein it should be noticed that new quantities are expressed in terms of the old, as regards the latter set, and not vice versa." Ex. gr. The symbols - dy, d z, ... are contragredient with the d- variables x, y, z, ... for when ( x , z, ���) = (A l, �i, VI I ���) (X, Y, Z, ���), I A 2, / 2 2, Y2, ... I I A S, 1 2 3, Y 3, .... 1 (Tr (T d d d d d d ,.. rd Y' ' ...) = 01, A2, A 3, ...) (d ' ' z / 2 1, /22, / 1 3, ... Pl, P2, P3, ...

Observe the notation, which is that introduced by Cayley into the theory of matrices which he himself created.

Just as cogrediency leads to a theory of covariants, so contragrediency leads to a theory of contravariants. If u, a quantic in x, y, z, ..., be expressed in terms of new variables X, Y, Z ...; and if, n,, ..., be quantities contragredient to x, y, z, ...; there are found to exist functions of, n, ?, ..., and of the coefficients in u, which need, at most, be multiplied by powers of the modulus to be made equal to the same functions of E, H, Z, ... of the transformed coefficients of u; such functions are called contravariants of u. There also exist functions, which involve both sets of variables as well as the coefficients of u, possessing a like property; such have been termed mixed concomitants, and they, like contravariants, may appertain as well to a system of forms as to a single form.

As between the original and transformed quantic we have the umbral relations A1 = A1a1 d-A2a2, A2 = /21a1+/22a2, and for a second form B1 =A 1 b 1+ A 2 b 2, B 2 =/21bl +�2b2� The original forms are ax, bi, and we may regard them either as different forms or as equivalent representations of the same form. In other words, B, b may be regarded as different or alternative symbols to A, a. In either case (AB) =A 1 B 2 -A 2 B 1 = (A/2)(ab); and, from the definition, (ab) possesses the invariant property. We cannot, however, say that it is an invariant unless it is expressible in terms of the real coefficients. Since (ab) = a l b 2 -a 2 b l, that this may be the case each form must be linear; and if the forms be different (ab) is an invariant (simultaneous) of the two forms, its real expression being aob l -a l b 0. This will be recognized as the resultant of the two linear forms. If the two linear forms be identical, the umbral sets a l, a2; b l, b 2 are alternative, are ultimately put equal to one another and (ab) vanishes. A single linear form has, in fact, no invariant. When either of the forms is of an order higher than the first (ab), as not being expressible in terms of the actual coefficients of the forms, is not an invariant and has no significance. Introducing now other sets of symbols C, D, ...; c, d, ... we may write (AB)i(AC)j(BC)k... _ (AIt)i+j+k+... (ab)i(ac)j(bc)k..., that the symbolic product (ab)i(ac)j(bc)k..., possesses the invariant property. If the forms be all linear and different, the function is an invariant, viz. the i t " power of that appertaining to a x and b x multiplied by the j t " power of that appertaining to a x and c x multiplied by &c. If any two of the linear forms, say p x, qx, be supposed identical, any symbolic expression involving the factor (pq) is zero. Notice, therefore, that the symbolic product (ab)i(ac)j(bc)k... may be always viewed as a simultaneous invariant of a number of different linear forms a x, x, c x, .... In order that (ab)i(ac)j(bc)k... may be a simultaneous invariant of a number of different forms az', bx 2, cx 3 ,..., where n1, n 2 , n3, ... may be the same or different, it is necessary that every product of umbrae which arises in the expansion of the symbolic product be of degree n, in a l, a 2; in the case of b,, b 2 of degree n 2; in the case of c 1, c 2 of degree n3; and so on. For these only will the symbolic product be replaceable by a linear function of products of real coefficients. Hence the condition is i+k+... =n2, j+k+... =n3, 'If' the forms a:, b:, cy 7 ...be identical the symbols are alternative, and provided that the form does not vanish it denotes an invariant of the single form ay. There may be a number of forms ay,bi,ci,... and we may suppose such identities between the symbols that on the whole only two, three, or more of the sets of umbrae are not equivalent; we will then obtain invariants of two, three, or more sets of binary forms. The symbolic expression of a covariant is equally simple, because we see at once that since AE, B, Ce,... are equal to a x, x, c x, ... respectively, the linear forms a x, b., cg, ... possess the invariant property, and we may write (AB) i (AC)'(BC) k ...A P E B C...

= t) 1 v ...axbxcx..., and assert that the symbolic product (ab)i(ac)'(bc)k...aibxc2... possesses the invariant property. It is always an invariant or covariant appertaining to a number of different linear forms, and as before it may vanish if two such linear forms be identical. In general it will be simultaneous covariant of the different forms n 1 rz 2 n3 a, b x, ? if i+j ?- ... +P=n1, j+k+...+T =n3, It will also be a covariant if the symbolic product be factorizable into portions each of which satisfies these conditions. If the forms be identical the sets of symbols are ultimately equated, and the form, provided it does not vanish, is a covariant of the form ate. The expression (ab) 4 properly appertains to a quartic; for a quadratic it may also be written (ab) 2 (cd) 2 , and would denote the square of the discriminant to a factor pres. For the quartic (ab) 4 = (aib2-a2b,) alb2 -4a7a2blb2+64a2 bib2 - 4a 1 a 2 b 7 b 2 + a a b i = a,a 4 - 4ca,a 3 +6a2 - 4a3a3+ aoa4 = 2(a 0 a 4 - 4a1a3 +e3a2), one of the well-known invariants of the quartic.

For the cubic (ab) 2 axbx is a covariant because each symbol a, b occurs three times; we can first of all find its real expression as a simultaneous covariant of two cubics, and then, by supposing the two cubics to merge into identity, find the expression of the quadratic covariant, of the single cubic, commonly known as the Hessian. By simple multiplication (al b l b2 -24a2bib2+ala2b;)xi +(aibz -ala214b2-aia2blb2+a2b2)xlx2 + (aia 2 b2 - 2a l a2b l b2 +a2/4b 2)x2; and transforming to the real form, (aob 2 - 2a1b,+a2bo)xi (aob 3 -a l b 2 - alb,+a3bo)xlx2 + (aib3 - 2a2b2+a3b1)x2, the simultaneous covariant; and now, putting b = a, we obtain twice. the Hessian ( 0 a 2 -al)xi + (a 0 a 3 -ala2)xlx2+ (a l a i - a2)x2. It will be shown later that all invariants, single or simultaneous, are expressible in terms of symbolic products. The degree of the covariant in the coefficients is equal to the number of different symbols a, b, c, ... that occur in the symbolic expression; the degree in the variables (i.e. the order of the covariant) is P+P+T+... and the weight' of the coefficient of the leading term xi +Q+T+.�� is equal to i+j+k+.... It will be apparent that there are four numbers associated with a covariant, viz. the orders of the quantic and covariant, and the degree and weight of the leading coefficient; calling these 'n, e,' 0, w respectively we can see that they are not independent integers, but that they are invariably connected by a certain relation n9 -2w = e. For, if c(ao i ...x l, x 2) be a covariant of order e appertaining to a quantic of order n, t (T. 0, ��. 1 2) = (A /-?) ' (ao,... A 1 1+/ 2 12, A 2E1 +/ 2 2 2) we find that the leftand right-hand sides are of degrees nO and 2w+e respectively in A,, �l, A 2, /22, and thence nO = 2w �E.

Symbolic Identities.- For the purpose of manipulating symbolic expressions it is necessary to be in possession of certain simple identities which connect certain symbolic products. From the three equations ax = alxl+ a2x2, b.= blxl+b2x2, cx = clxi+c2x2, we find by eliminating x, and x 2 the relation a x (bc)+b x (ca) +c x (ab) =0. .. (I.) Introduce now new umbrae dl, d 2 and recall that +d 2 -d 1 are cogredient with x, and x 2. We may in any relation substitute for any pair of quantities any other cogredient pair so that writing -}-d 2, -d l for x 1 and x 2, and noting that gx then becomes (gd), the above-written identity bceomes (ad)(bc)+(bd)(ca)+(cd)(ab) = 0. .. (II.) Similarly in (I.), writing for c l, c 2 the cogredicnt pair -y2, +y1, we obtain axb5-a5bx=(ab)(xy).. -.. . (III.) Again in (I.) transposing a x (bc) to the other side and squaring, we obtain 2(ac) (bc)axbx = (bc) 2 a'+(ac) 2 bx- (ab) 2 c1. (IV.) and herein writing d 2, -d 1 for x l, x2, 2 (ac) (bc) (ad) (bd) = (bc) 2 (ad) 2 +(ac) 2 (bd) 2 - (ab) 2 (cd) 2. (V.) As an illustration multiply (IV.) throughout by az 2b x 2cz 2 so that each term may denote a covariant of an ni°. 2 (ac)(bc)anx xibn-i -1 x = (bc)2anbn-2Cn-2 + (ac)2an x x x The weight of a term ao°a l l ...an n is defined as being k,+2k2+...

+nkn. -2 _ ab 2an-2bn-2Crz z x () x x x, Each term on the right-hand side may be shown by permutation of a, b, c to be the symbolical representation of the same covariant; they are equivalent symbolic products, and we may accordingly write 2(ac) (bc)ai -1 bi -1 cx 2 =(ab)2a:-2b:-2c:, a relation which shows that the form on the left is the product of the two covariants n (ab) ay 2 by 2 and cZ.

The identities are, in particular, of service in reducing symbolic products to standard forms. A symbolical expression may be always so transformed that the power of any determinant factor (ab) is even. For we may in any product interchange a and b without altering its signification; therefore (ab) 2m+1 4) 1 = - (ab) 2 " 4)2, where 4,1 becomes by the interchange, and hence (ab)2m+14)1= Z (ab) 2m+1 (4) 1 - 02); and identity (I.) will always result in transforming 01-02 so as to make it divisible by (ab).

Ex. gr. (ab)(ac)bxcx = - (ab)(bc)axcx = 2(ab)c x {(ac)bx-(bc)axi = 1(ab)2ci; so that the covariant of the quadratic on the left is half the product of the quadratic itself and its only invariant. To obtain the corresponding theorem concerning the general form of even order we multiply throughout by (ab)2' 2c272 and obtain (ab)2m-1(ac)bxc2:^1=(ab)2mc2 Paying attention merely to the determinant factors there is no form with one factor since (ab) vanishes identically. For two factors the standard form is (ab) 2; for three factors (ab) 2 (ac); for four factors (ab) 4 and (ab) 2 (cd) 2; for five factors (ab) 4 (ac) and (ab) 2 (ac)(de) 2; for six factors (ab) 6, (ab) 2 (bc) 2 (ca) 2 , and (ab) 2 (cd) 2 (ef) 2 . It will be a useful exercise for the reader to interpret the corresponding covariants of the general quantic, to show that some of them are simple powers or products of other covariants of lower degrees and order.

The Polar Process

The �th polar of ax with regard to y is n-� a aye i.e. of the symbolic factors of the form are replaced by IA others in which new variables y1, y2 replace the old variables x1, x 2 . The operation of taking the polar results in a symbolic product, and the repetition of the process in regard to new cogredient sets of variables results in symbolic forms. It is therefore an invariant process. All the forms obtained are invariants in regard to linear transformations, in accordance with the same scheme of substitutions, of the several sets of variables.

An important associated operation is a ? 32 ax l ay 2 ax2ay1' which, operating upon any polar, causes it to vanish. Moreover, its operation upon any invariant form produces an invariant form. Every symbolic product, involving several sets of cogredient variables, can be exhibited as a sum of terms, each of which is a polar multiplied by a product of powers of the determinant factors ( xy), (xz), (yz),... Transvection. - We have seen that (ab) is a simultaneous invariant of the two different linear forms a x, bx, and we observe that (ab) is equivalent to where f =a x, 4)=b. If f =ay, 4 = b' be any two binary forms, we generalize by forming the function (m-k)! (n-k)! of a4) of a 4) k m! l ax 2 2 ax i l This is called the kth transvectant of f over 4); it may be conveniently denoted by (f, (15)k. (a m b n) k (ab) kamkbn-k x, x - x it is clear that the k th transvectant is a simultaneous covariant of the two forms.

It has been shown by Gordan that every symbolic product is expressible as a sum of transvectants.

If m > n there are n +1 transvectants corresponding to the values o, t, 2,... n of k; if k = o we have the product of the two forms, and for all values of k>n the transvectants vanish. In general we may have any two forms 01/1X1+ 'II � Yy + 02x2) p Y'x =, / / being the umbrae, as usual, and for the kth transvectant we have (4)1,,, 4)Q) k = (4)) k 4)2 -krk, a simultaneous covariant of the two forms. We may suppose of, 4 ,2 to be any two covariants appertaining to a system, and the process of transvection supplies a means of proceeding from them to other covariants.

The two forms ax, bx, or of, 0, may be identical; we then have the kth transvectant of a form over itself which may, or may not, vanish identically; and, in the latter case, is a covariant of the single form. It is obvious that, when k is uneven, the kth transvectant of a form over itself does vanish. We have seen that transvection is equivalent to the performance of partial differential operations upon the two forms, but, practically, we may regard the process as merely substituting (ab) k, (OW for azbx, 4x t ' respectively in the symbolic product subjected to transvection. It is essentially an operation performed upon the product of �two forms. If, then, we require the transvectants of the two forms f+Xf', 0+14', we take their product fc5+xf'95+,-ifct'+atif'cb', and the kth transvectant is simply obtained by operating upon each term separately, viz.

(f, 4)) k +(f, 4)) k +�(f, 4/) k +a�(1, 4)')k; and, moreover, if we require to find the kth transvectant of one linear system of forms over another we have merely to multiply the two systems, and take the k th transvectant of the separate products.

The process of transvection is connected with the operations 12; for ?k (a m b n) = (ab)kam-kbn-k, (x y x y or S 2 k (a x by) x = 4))k; so also is the polar process, for since f k m-k k k n - k k y = a x by, 4)y = bx by, if we take the k th transvectant of f i x; over 4 k, regarding y,, y 2 as the variables, (f k, 4)y) k (ab) ka x -kb k (f, 15)k; or the k th transvectant of the k th polars, in regard to y, is equal to the kth transvectant of the forms. Moreover, the kth transvectant (ab) k a m-k b: -k is derivable from the kth polar of ax, viz. ai by substituting for y 1, y 2 the cogredient quantities b2,-b1, and multiplying by by-k. First and Second Transvectants. - A few words must be said about the first two transvectants as they are of exceptional Interest. Since, If F = An, 4) = By, 1 = I (Df A4) Of A?) Ab A"'^1Bz 1=, (F, Mn Ax I Ax 2 Axe Ax1) J The First Transvectant Differs But By A Numerical Factor From The Jacobian Or Functional Determinant, Of The Two Forms. We Can Find An Expression For The First Transvectant Of (F, �) 1 Over Another Form Cp. For (M N)(F,4)), =Nf.4Y Mfy.4), And F,4, F 5.4)= (Axby A Y B X) A X B X 1= (Xy)(F,4))1; (F,Ct)1=F5.D' 7,(Xy)(F4)1. Put M 1 For M, N I For N, And Multiply Through By (Ab); Then { (F ,C6) } = (Ab) A X 2A Y B X 1 M N I 2 (Xy) ,?) 2, = (A B)Ax 1B X 2B Y L I Multiply By Cp 1 And For Y L, Y2 Write C 2, C1; Then The Right Hand Side Becomes (Ab)(Bc)Am Lbn 2Cp 1 M I C P (F?) 2 M { N2 X, Of Which The First Term, Writing C P = ,,T, Is Mn 2 A B (Ab)(Bc)Axcx 1 M 2 N 2 P 2 2222 2 2 _2 A X B X C (Bc) A C Bx M N 2 2 2 M2°N 2 N 2 M 2 2 A X (Bc) B C P C P (Ab) A B B(Ac) Ax Cp 2 = 2 (04) 2 1 (F,0) 2.4 (F,Y') 2 �?; And, If (F,4)) 1 = Km " 2, (F??) 1 1 M N S X X X Af A _Af A Ax, Ax Ax Ax1 Observing That And This, On Writing C 2, C 1 For Y 11 Y 21 Becomes ( Kc) K X 'T 3C X 1= (F,0 1 ', G 1; �'�1(F,O) 1 M 1=1 M 2 0`,4)) 2 0, T (Fm 2.4 (0,0 2 .F ' And Thence It Appears That The First Transvectant Of (F, (P) 1 Over 4) Is Always Expressible By Means Of Forms Of Lower Degree In The Coefficients Wherever Each Of The Forms F, 0, 4, Is Of Higher Degree Than The First In X 1, X2.

The second transvectant of a form over itself is called the Hessian of the form. It is (f = (ab ) 2 a n-2 r7 2 =Hx - =H; unsymbolically bolically it is a numerical multiple of the determinant a2 f a2f (32 f) 2� It is also the first transvectant of the differxi ax axa x 2 ential coefficients of the form with regard to the variables, viz. (L, _f_)'. For the quadratic it is the discriminant (ab) 2 and for ax2 the cubic the quadratic covariant (ab) 2 axbx. In general for a form in n variables the Hessian is 3 2 f 3 2 f a2f ax i ax n ax 2 ax " �� ' axn and there is a remarkable theorem which states that if H =o and n=2, 3, or 4 the original form can be exhibited as a form in I, 2, 3 variables respectively.

The Form f+A4. - An important method for the formation of covariants is connected with the form f +X4), where f and 4 are of the same order in the variables and X is an arbitrary constant. If the invariants and covariants of this composite quantic be formed we obtain functions of X such that the coefficients of the various powers of X are simultaneous invariants of f and 4). In particular, when 4) is a covariant of f, we obtain in this manner covariants of f. The Partial Differential Equations.--It will be shown later that covariants may be studied by restricting attention to the leading coefficient, viz. that affecting xi where e is the order of the covariant.

An important fact, discovered by Cayley, is that these coefficients, and also the complete covariants, satisfy certain partial differential equations which suffice to determine them, and to ascertain many of their properties. These equations can be arrived at in many ways; the method here given is due to Gordan. X1, X 2, u1, /22 being as usual the coefficients of substitution, let x1a ? + X 2 - = D, X 1 -' j +X 2 =D 2 AA' ?2 / 2 1 3 - 5 -, =112 87,2 = ?1a a + ?2a a =D��, 1 be linear operators. Then if j, J be the original and transformed forms of an invariant J= (a1)wj, w being the weight of the invariant.

Operation upon J results as follows D AA J = wJ; D A J=0; D �A J =0;D �� J = wJ.

The first and fourth of these indicate that (a 2) w is a homogeneous function of X i, X2, and of /u1, � 2 separately, and the second and third arise from the fact that (X / 1) is caused to vanish by both Da � and D�A. Since J= F(A0,A11...Ak,�..), where A k= we find that the results are equivalent to. aJ - ., _ A aJ �. k (DwAk) Ak 0; (D (� A k) Ak =wJ.

k k According to the well-known law for the changes of independent variables. Now D A xA k = (n - k) A k; A� A k = k A?1; D �A A k = (n - k) A k+1;D m� A k = kA k; (n - k)A ka - w Ak - 1 aA k = O; a _ J (n - k) A k +l A k = O; kA k Ak = wJ; equations which are valid when X 1, X 2, � 1, �2 have arbitrary values, and therefore when the values are such that J =j, A k =ak� Hence °a-do +(n -1)71 (a2aa-+... =wj, - aj aj - aj a °aa1 +2a 1aa2 +3a 2aa3 +... =0, - aj aj aj nal aao +(n-1)a2 at i -} (n - 2)a 3aa2+... =0, a 1 a ? +2a 2 a? +3a 3 a +... = wj, aa 1 aa 2 a a 3 the complete system of equations satisfied by an invariant. The fourth shows that every term of the invariant is of the same weight. Moreover, if we add the first to the fourth we obtain aj 2w ak = 7 1=6, j, =0j, where 0 is the degree of the invariant; this shows, as we have before observed, that for an invariant w= - n0. The second and third are those upon the solution of which the theory of the invariant may be said to depend. An instantaneous deduction from the relation w= 2 n0 is that forms of uneven orders possess only invariants of even degree in the coefficients. The two operators - a a - a = a °aa 1 +2 a 1aa2 +... +na" -laan -a a O = na laao + (n 1)a 2aa1 +�.. +a"aa"-1 have been much studied by Sylvester, Hammond, Hilbert and Elliott (Elliott, Algebra of Quantics, ch. vi.). An important reference is " The Differential Equations satisfied by Concomitants of Quantics," by A. R. Forsyth, Proc. Lond. Math. Soc. vol. xix.

The Evectant Process

If we have a symbolic product, which contains the symbol a only in determinant factors such as (ab), we may write x 2 ,-x 1 for a 1, a 2 , and thus obtain a product in which (ab) is replaced by b x, (ac) by c x and so on. In particular, when the product denotes an invariant we may transform each of the symbols a, b,...to x in succession, and take the sum of the resultant products; we thus obtain a covariant which is called the first evectant of the original invariant. The second evectant is obtained by similarly operating upon all the symbols remaining which only occur in determinant factors, and so on for the higher evectants.

Ex. gr. From (ac) 2 (bd) 2 (ad)(bc) we obtain (bd) 2 (bc) cyd x +(ac) 2 (ad) c xdx - (bd) 2 (ad)axb x - (ac)2(bc)axbx =4(bd) 2 (bc)c 2. d x the first evectant; and thence 4cxdi the second evectant; in fact the two evectants are to numerical factors pres, the cubic covariant Q, and the square of the original cubic.

If 0 be the degree of an invariant j - aj aj a; oj =a ° a a o +al aa l +... +anaan naj n.-1 aj naj =a l aa ° +a 1 a2c3a1...+a2aan and, herein transforming from a to x, we obtain the first evectant (-) k, x1x2 aak k Combinants. - An important class of invariants, of several binary forms of the same order, was discovered by Sylvester. The invariants in question are invariants qud linear transformation of the forms themselves as well as qud linear transformation of the variables.

If the forms be ax, b2, cy,... The Aronhold process, given by the operation a as between any two of the forms, causes such an invariant to vanish. Thus it has annihilators of the forms a0 db - 0 +al d 1+a2d 22+... °c - iao l a12da2+'.. and Gordan, in fact, takes the satisfaction of these conditions as defining those invariants which Sylvester termed " combinants." The existence of such forms seems to have been brought to Sylvester's notice by observation of the fact that the resultant of of and b must be a factor of the resultant of Xax+ 12 by and X'a +tA2 for a common factor of the first pair must be also a common factor so we obtain P: = of the second pair; so that the condition for the existence of such common factor must be the same in the two cases. A leading proposition states that, if an invariant of Xax and i ubi be considered as a form in the variables X and ,u, and an invariant of the latter be taken, the result will be a combinant of cif and b1'. The idea_can be generalized so as to have regard to ternary and higher forms each of the same order and of the same number of variables.

For further information see Gordan, Vorlesungen Tiber Invariantentheorie, Bd. ii. � 6 (Leipzig, 1887); E. B. Elliott, Algebra of Quantics, Art. 264 (Oxford, 1895).

Associated Forms.-A system of forms, such that every form appertaining to the binary form is expressible as a rational and integral function of the members of the system, is difficult to obtain. If, however, we specify that all forms are to be rational, but not necessarily integral functions, a new system of forms arises which is easily obtainable. A binary form of order n contains n independent constants, three of which by linear transformation can be given determinate values; the remaining n-3 coefficients, together with the determinant of transformation, give us n -2 parameters, and in consequence one relation must exist between any n - I invariants of the form, and fixing upon n-2 invariants every other invariant is a rational function of its members. Similarly regarding 1 x 2 as additional parameters, we see that every covariant is expressible as a rational function of n fixed covariants. We can so determine these n covariants that every other covariant is expressed in terms of them by a fraction whose denominator is a power of the binary form.

First observe that with f x =a: = b z = ���,f1 = a l a z ',