1911 Encyclopædia Britannica/Algebraic Forms

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 1773 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. 80-94, 95-112) 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.[1]

I. The Theory of Determinants.[1]

Let there be given ${\displaystyle n^{2}}$ quantities

${\displaystyle {\begin{matrix}a_{11}&a_{12}&a_{13}&...&a_{1n}\\a_{21}&a_{22}&a_{23}&...&a_{2n}\\a_{31}&a_{32}&a_{33}&...&a_{3n}\\.&.&.&...&.\\a_{n1}&a_{n2}&a_{n3}&...&a_{nn}\end{matrix}}}$

and form from them a product of ${\displaystyle n}$ quantities

${\displaystyle {\begin{matrix}a_{1\alpha }&a_{2\beta }&a_{3\gamma }&...&a_{n\nu },\end{matrix}}}$

where the first suffixes are the natural numbers ${\displaystyle 1,2,3,...n}$ taken in order, and ${\displaystyle \alpha ,\beta ,\gamma ,...\nu }$ is some permutation of these ${\displaystyle n}$ numbers. This permutation by a transposition of two numbers, say ${\displaystyle \alpha ,\beta ,}$ becomes ${\displaystyle \beta ,\alpha ,\gamma ,...\nu ,}$ and by successively transposing pairs of letters the permutation can be reduced to the form ${\displaystyle 1,2,3,...n}$. Let ${\displaystyle k}$ such transpositions be necessary; then the expression

${\displaystyle \Sigma (-)^{k}a_{1\alpha }a_{2\beta }a_{3\gamma }...a_{n\nu }}$,

the summation being for all permutations of the ${\displaystyle n}$ numbers, is called the determinant of the ${\displaystyle n^{2}}$ quantities. The quantities ${\displaystyle a_{1\alpha }a_{2\beta }...}$ are called the elements of the determinant; the term ${\displaystyle (-)^{k}a_{1\alpha }a_{2\beta }a_{3\gamma }...a_{n\nu }}$ is called a member of the determinant, and there are evidently ${\displaystyle n!}$ members corresponding to the ${\displaystyle n!}$ permutations of the ${\displaystyle n}$ numbers ${\displaystyle 1,2,3,...n}$. The determinant is usually written

${\displaystyle \Delta ={\begin{vmatrix}a_{11}&a_{12}&a_{13}&...&a_{1n}\\a_{21}&a_{22}&a_{23}&...&a_{2n}\\a_{31}&a_{32}&a_{33}&...&a_{3n}\\.&.&.&...&.\\a_{n1}&a_{n2}&a_{n3}&...&a_{nn}\end{vmatrix}}}$

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 ${\displaystyle n}$ rows and ${\displaystyle n}$ 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.

Theorem.—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 ${\displaystyle a_{11}a_{22}a_{33}...a_{nn}}$, and corresponds to the principal diagonal of the matrix.

We write frequently

${\displaystyle \Delta =\Sigma \pm a_{11}a_{22}a_{33}...a_{nn}=(a_{11}a_{22}a_{33}...a_{nn}).}$

If the first two columns of the determinant be transposed the expression for the determinant becomes ${\displaystyle \Sigma (-)^{k}a_{1\beta }a_{2\alpha }a_{3\gamma }...a_{n\nu }}$, viz. ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are transposed, and it is clear that the number of transpositions necessary to convert the permutation ${\displaystyle \beta \alpha \gamma ...\nu }$ 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.

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

Corollary.—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 ${\displaystyle \Delta }$ we may separate those members which contain a particular element ${\displaystyle a_{ik}}$ as a factor, and write the portion ${\displaystyle a_{ik}{\text{A}}_{ik}}$; ${\displaystyle {\text{A}}_{ik}}$, the cofactor of ${\displaystyle a_{ik}}$, is called a minor of order ${\displaystyle n-1}$ of the determinant.

Now ${\displaystyle a_{11}{\text{A}}_{11}=\Sigma \pm a_{11}a_{22}a_{33}...a_{nn}}$, wherein ${\displaystyle a_{11}}$ is not to be changed, but the second suffixes in the product ${\displaystyle a_{22}a_{33}...a_{nn}}$ assume all permutations, the number of transpositions necessary determining the sign to be affixed to the member.

Hence ${\displaystyle a_{11}{\text{A}}_{11}=a_{11}\Sigma \pm a_{22}a_{33}...a_{nn}}$, where the cofactor of ${\displaystyle a_{11}}$ is clearly the determinant obtained by erasing the first row and the first column.

Hence ${\displaystyle {\text{A}}_{11}={\begin{vmatrix}a_{22}&a_{23}&...&a_{2n}\\a_{32}&a_{33}&...&a_{3n}\\.&.&...&.\\a_{n2}&a_{n3}&...&a_{nn}\end{vmatrix}}}$

Similarly ${\displaystyle {\text{A}}_{ik}}$, the cofactor of ${\displaystyle a_{ik}}$, is shown to be the product of ${\displaystyle (-)^{ik}}$ and the determinant obtained by erasing from ${\displaystyle \Delta }$ the i th 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

${\displaystyle \Delta =a_{11}{\text{A}}_{11}+a_{12}{\text{A}}_{12}+a_{13}{\text{A}}_{13}+...+a_{1n}{\text{A}}_{1n}}$,

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.

 ${\displaystyle \Delta =a_{i1}{\text{A}}_{i1}+a_{i2}{\text{A}}_{i2}+a_{i3}{\text{A}}_{i3}+...+a_{in}{\text{A}}_{in}}$ ${\displaystyle {\Bigg \rbrace }=({\text{A}}).}$ ${\displaystyle \Delta =a_{1k}{\text{A}}_{1k}+a_{2k}{\text{A}}_{2k}+a_{3k}{\text{A}}_{3k}+...+a_{nk}{\text{A}}_{nk}}$

This theory enables the evaluation of a determinant by successive reduction of the orders of the determinants involved.

 Ex. gr. ${\displaystyle {\begin{vmatrix}1&0&3\\2&1&6\\0&-5&3\end{vmatrix}}}$ ${\displaystyle =}$ ${\displaystyle 1{\begin{vmatrix}1&6\\-5&3\end{vmatrix}}-0{\begin{vmatrix}2&6\\0&3\end{vmatrix}}+3{\begin{vmatrix}2&1\\0&-5\end{vmatrix}}}$ ${\displaystyle =}$ ${\displaystyle 1\left|3\right|-6\left|-5\right|+3.2\left|-5\right|-3.1\left|0\right|}$ ${\displaystyle =}$ ${\displaystyle 3+30-30-0=3.}$

Since the determinant

${\displaystyle {\begin{vmatrix}a_{11}&a_{22}&a_{23}&...&a_{2n}\\a_{11}&a_{22}&a_{23}&...&a_{2n}\\a_{31}&a_{32}&a_{33}&...&a_{3n}\\.&.&.&...&.\\a_{n1}&a_{n2}&a_{n3}&...&a_{nn}\end{vmatrix}}}$, having two identical rows,

vanishes identically; we have by development according to the elements of the first row

${\displaystyle a_{21}{\text{A}}_{11}+a_{22}{\text{A}}_{12}+a_{23}{\text{A}}_{13}+...+a_{2n}{\text{A}}_{1n}=0}$;

and, in general, since

${\displaystyle a_{i1}{\text{A}}_{i1}+a_{i2}{\text{A}}_{i2}+a_{i3}{\text{A}}_{i3}+...+a_{in}{\text{A}}_{in}=\Delta }$,

if we suppose the ith and kth rows identical

${\displaystyle a_{k1}{\text{A}}_{i1}+a_{k2}{\text{A}}_{i2}+a_{k3}{\text{A}}_{i3}+...+a_{kn}{\text{A}}_{in}=0\qquad (k\gtrless i)}$;

and proceeding by columns instead of rows,

${\displaystyle a_{1i}{\text{A}}_{1k}+a_{2i}{\text{A}}_{2k}+a_{3i}{\text{A}}_{3k}+...+a_{ni}{\text{A}}_{nk}=0\qquad (k\gtrless i)}$

identical relations always satisfied by these minors.

If in the first relation of ${\displaystyle (A)}$ we write ${\displaystyle a_{is}=b_{is}+c_{is}+d_{is}+...}$ we find that ${\displaystyle \Sigma a_{is}{\text{A}}_{is}=\Sigma b_{is}{\text{A}}_{is}+\Sigma c_{is}{\text{A}}_{is}+\Sigma d_{is}{\text{A}}_{is}+...}$ so that ${\displaystyle \Delta }$ breaks up into a sum of determinants, and we also obtain a theorem for the addition of determinants which have ${\displaystyle n-1}$ rows in common. If we multiply the elements of the second row by an arbitrary magnitude ${\displaystyle \lambda }$, and add to the corresponding elements of the first row, ${\displaystyle \Delta }$ becomes ${\displaystyle \Sigma a_{1s}{\text{A}}_{1s}+\lambda \Sigma a_{2s}{\text{A}}_{1s}=\Delta }$, 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.

Observation.—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. ${\displaystyle {\begin{vmatrix}\alpha ^{2}&\beta ^{2}&\gamma ^{2}\\\alpha &\beta &\gamma \\1&1&1\end{vmatrix}}={\begin{vmatrix}\alpha ^{2}&\beta ^{2}-\alpha ^{2}&\gamma ^{2}-\alpha ^{2}\\\alpha &\beta -\alpha &\gamma -\alpha \\1&0&0\end{vmatrix}}={\begin{vmatrix}\beta ^{2}-\alpha 2&\gamma ^{2}-\alpha ^{2}\\\beta -\alpha &\gamma -\alpha \end{vmatrix}}}$ ${\displaystyle =(\beta -\alpha )(\gamma -\alpha ){\begin{vmatrix}\beta +\alpha &\gamma +\alpha \\1&1\end{vmatrix}}=(\beta -\gamma )(\gamma -\alpha ){\begin{vmatrix}\beta -\gamma &\gamma +\alpha \\0&1\end{vmatrix}}}$ ${\displaystyle =(\beta -\alpha )(\gamma -\alpha )(\beta -\gamma ).}$

The minor ${\displaystyle {\text{A}}_{ik}}$ is ${\displaystyle {\frac {\partial \Delta }{\partial a_{ik}}}}$, and is itself a determinant of order ${\displaystyle n-1}$. We may therefore differentiate again in regard to any element ${\displaystyle a_{rs}}$ where ${\displaystyle r\gtrless i}$, ${\displaystyle s\gtrless k}$; we will thus obtain a minor of ${\displaystyle {\text{A}}_{ik}}$, which is a minor also of ${\displaystyle \Delta }$ of order ${\displaystyle n-2}$. It will be ${\displaystyle {\text{A}}_{ik \atop rs}={\frac {\partial {\text{A}}_{ik}}{\partial a_{rs}}}={\frac {\partial ^{2}\Delta }{\partial a_{ik}\partial a_{rs}}}}$ and will be obtained by erasing from the determinant ${\displaystyle {\text{A}}_{ik}}$ the row and column containing the element ${\displaystyle a_{rs}}$; this was originally the r th row and the sth column of ${\displaystyle \Delta }$; the r th row of ${\displaystyle \Delta }$ is the r th or (r–1)th row of ${\displaystyle {\text{A}}_{ik}}$ according as ${\displaystyle r\gtrless i}$ and the sth column of ${\displaystyle \Delta }$ is the sth or (s−1)th column of ${\displaystyle {\text{A}}_{ik}}$ according as ${\displaystyle s\gtrless k}$. Hence, if ${\displaystyle T_{ri}}$ denote the number of transpositions necessary to bring the succession ${\displaystyle ri}$ into ascending order of magnitude, the sign to be attached to the determinant arrived at by erasing the ith and r th rows and the k th and s th columns from ${\displaystyle \Delta }$ in order produce ${\displaystyle {\text{A}}_{ik \atop rs}}$ will be ${\displaystyle -1}$ raised to the power of ${\displaystyle T_{ri}+T_{ks}+i+k+r+s}$.

Similarly proceeding to the minors of order ${\displaystyle n-3}$, we find that ${\displaystyle {\text{A}}_{ik \atop {rs \atop tu}}={\frac {\partial }{\partial a_{tu}}}{\text{A}}_{ik \atop rs}={\frac {\partial ^{2}}{\partial a_{rs}\partial a_{tu}}}{\text{A}}_{ik}={\frac {\partial ^{3}}{\partial a_{ik}\partial a_{rs}\partial a_{tu}}}\Delta }$ is obtained from ${\displaystyle \Delta }$ by erasing the i th, r th, t th, rows, the k th, s th, u th columns, and multiplying the resulting determinant by ${\displaystyle -1}$ raised to the power ${\displaystyle T_{tri}+T_{usk}+i+k+r+s+t+u}$ and the general law is clear.

Corresponding Minors.—In obtaining the minor ${\displaystyle {\text{A}}_{ik \atop rs}}$ 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 ${\displaystyle {\text{A}}_{is \atop rk}}$, since the deleting lines intersect in two pairs of points. In the latter case the sign is determined by ${\displaystyle -1}$ raised to the same power as before, with the exception that ${\displaystyle T_{uks}}$, replaces ${\displaystyle T_{usk}}$; but if one of these numbers be even the other must be uneven; hence

${\displaystyle {\text{A}}_{ik \atop rs}=-{\text{A}}_{is \atop rk}}$.

Moreover

${\displaystyle a_{ik}a_{rs}{\text{A}}_{is \atop rk}={\begin{vmatrix}a_{ik}&a_{is}\\a_{ik}&a_{rs}\end{vmatrix}}{\text{A}}_{ik \atop rs}}$,

where the determinant factor is given by the four points in which the deleting lines intersect. This determinant and that associated with ${\displaystyle {\text{A}}_{ik \atop rs}}$ are termed corresponding determinants. Similarly ${\displaystyle p}$ lines of deletion intersecting in ${\displaystyle p^{2}}$ points yield corresponding determinants of orders ${\displaystyle p}$ and ${\displaystyle n-p}$ respectively. Recalling the formula

${\displaystyle \Delta =a_{11}{\text{A}}_{11}+a_{12}{\text{A}}_{12}+a_{13}{\text{A}}_{13}+...+a_{1n}{\text{A}}_{1n}}$,

it will be seen that ${\displaystyle a_{1k}}$ and ${\displaystyle {\text{A}}_{1k}}$ involve corresponding determinants. Since ${\displaystyle {\text{A}}_{1k}}$ is a determinant we similarly obtain

${\displaystyle {\text{A}}_{1k}=a_{21}{\text{A}}_{1k \atop 21}+...+a_{2,k-1}{\text{A}}_{1,k \atop 2,k-1}+...+a_{2,n}{\text{A}}_{1,k \atop 2,n}}$,

and thence

${\displaystyle \Delta =\Sigma _{i,k}a_{1i}a_{2i}{\text{A}}_{1i \atop 2k}\quad i}$${\displaystyle k}$;

and as before

${\displaystyle \Delta =\sum _{i,k}{\begin{vmatrix}a_{1i}&a_{2i}\\a_{1k}a_{2k}\end{vmatrix}}{\text{A}}_{1i \atop 2k}\quad i>k}$,

an important expansion of ${\displaystyle \Delta }$.

Similarly

${\displaystyle \Delta =\sum _{i,k,r}{\begin{vmatrix}a_{1i}&a_{2i}&a_{3i}\\a_{1k}&a_{2k}&a_{3k}\\a_{1r}&a_{2r}&a_{3r}\end{vmatrix}}{\text{A}}_{1i \atop {2k \atop 3r}}\quad i>k>r}$,

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 ith the determinant ${\displaystyle \Delta }$ vanishes identically; hence if ${\displaystyle j}$ be not equal to ${\displaystyle i}$, ${\displaystyle k}$ or ${\displaystyle r}$,

${\displaystyle 0=\sum {\begin{vmatrix}a_{1j}&a_{2j}&a_{3j}\\a_{1k}&a_{2k}&a_{3k}\\a_{1r}&a_{2r}&a_{3r}\end{vmatrix}}{\text{A}}_{1i \atop {2k \atop 3r}}}$.

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 ${\displaystyle \Delta =(a_{11},a_{22},...a_{nn})}$ and ${\displaystyle D=(b_{11},b_{22},b_{nn})}$ may be written as a determinant of order ${\displaystyle 2n}$, viz.

${\displaystyle {\begin{vmatrix}a_{11}&a_{21}&a_{31}&...&a_{n1}&-1&0&0&...&0\\a_{12}&a_{22}&a_{32}&...&a_{n2}&0&-1&0&...&0\\a_{13}&a_{23}&a_{33}&...&a_{n3}&0&0&-1&...&0\\.&.&.&...&.&.&.&.&...&.\\a_{1n}&a_{2n}&a_{3n}&...&a_{nn}&0&0&0&...&-1\\0&0&0&...&0&b_{11}&b_{12}&b_{13}&...&b_{1n}\\0&0&0&...&0&b_{21}&b_{22}&b_{23}&...&b_{2n}\\0&0&0&...&0&b_{31}&b_{32}&b_{33}&...&b_{3n}\\0&0&0&...&0&b_{n1}&b_{n2}&b_{n3}&...&b_{nn}\\\end{vmatrix}}~{\begin{matrix}={\begin{vmatrix}{\text{A}}&{\text{B}}\\{\text{C}}&{\text{D}}\end{vmatrix}}\\{\text{for brevity.}}\end{matrix}}}$

Multiply the 1st, 2nd ... nth rows by ${\displaystyle b_{11},b_{12},...b_{1n}}$ respectively, and add to the (n+1)th row; by ${\displaystyle b_{21},b_{22}\dots b_{2n}}$, and add to the (n+2)th row; by ${\displaystyle b_{31},b_{32}\dots b_{3n}}$ and add to the (n+3)rd row, &c. C then becomes

${\displaystyle {\begin{vmatrix}a_{11}b_{11}+a_{12}b_{12}+\dots +a_{1n}b_{1n},&a_{21}b_{11}+a_{22}b_{12}+\dots +a_{2n}b_{1n},&\dots &a_{n1}b_{11}+a_{n2}b_{12}+\dots +a_{nn}b_{1n}\\a_{11}b_{21}+a_{12}b_{22}+\dots +a_{1n}b_{2n},&a_{21}b_{21}+a_{22}b_{22}+\dots +a_{2n}b_{2n},&\dots &a_{n1}b_{21}+a_{n2}b_{22}+\dots +a_{nn}b_{2n}\\a_{11}b_{31}+a_{12}b_{32}+\dots +a_{1n}b_{3n},&a_{21}b_{31}+a_{22}b_{32}+\dots +a_{2n}b_{3n},&\dots &a_{n1}b_{31}+a_{n2}b_{32}+\dots +a_{nn}b_{3n}\\.~~.~~.&.~~.~~.&&.~~.~~.\\a_{11}b_{n1}+a_{12}b_{n2}+\dots +a_{1n}b_{nn},&a_{21}b_{n1}+a_{22}b_{n2}+\dots +a_{2n}b_{nn},&\dots &a_{n1}b_{n1}+a_{n2}b_{n2}+\dots +a_{nn}b_{nn}\\\end{vmatrix}}}$

and all the elements of D become zero. Now by the expansion theorem the determinant becomes

${\displaystyle (-)^{1+2+3+\ldots +2n}{\mbox{B.C}}=(-1)^{n(2n+1)+n}{\mbox{C}}={\mbox{C}}.}$

We thus obtain for the product a determinant of order ${\displaystyle n}$. We may say that, in the resulting determinant, the element in the i th row and kth column is obtained by multiplying the elements in the kth row of the first determinant severally by the elements in the i th row of the second, and has the expression

${\displaystyle a_{k1}b_{i1}+a_{k2}b_{i2}+a_{k3}b_{i3}\dots +a_{kn}b_{in}}$

,

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 determinant of the same order ${\displaystyle (b_{11}b_{22}b_{33}\dotsb _{nn})}$ such that ${\displaystyle b_{ik}=b_{ki}}$; it is for this reason termed symmetrical.

The Adjoint or Reciprocal Determinant arises from ${\displaystyle \Delta =(a_{11}a_{22}a_{33}\dots a_{nn})}$ by substituting for each element ${\displaystyle {\mbox{A}}_{ik}}$ the corresponding minor ${\displaystyle {\mbox{A}}_{ik}}$ so as to form ${\displaystyle {\mbox{D}}=({\mbox{A}}_{11}{\mbox{A}}_{22}{\mbox{A}}_{33}\dots {\mbox{A}}_{nn})}$. If we form the product ${\displaystyle \Delta .{\mbox{D}}}$ by the theorem for the multiplication of determinants we find that the element in the i th row and kth column of the product is

${\displaystyle a_{ki}{\mbox{A}}_{i1}+a_{k2}{\mbox{A}}_{i2}+\dots +a_{kn}{\mbox{A}}_{in}}$,

the value of which is zero when ${\displaystyle k}$ is different from ${\displaystyle i}$, whilst it has the value ${\displaystyle \Delta }$ when ${\displaystyle k=i}$. Hence the product determinant has the principal diagonal elements each equal to ${\displaystyle \Delta }$ and the remaining elements zero. Its value is therefore ${\displaystyle \Delta ^{n}}$ and we have the identity

${\displaystyle {\mbox{D}}.\Delta =\Delta ^{n}}$ or ${\displaystyle {\mbox{D}}=\Delta ^{n-1}}$.

It can now be proved that the first minor of the adjoint determinant, say ${\displaystyle {\mbox{B}}_{rs}}$ is equal to ${\displaystyle \Delta ^{n-2}a_{rs}}$.

From the equations

 ${\displaystyle a_{11}x_{1}}$ ${\displaystyle +}$ ${\displaystyle a_{12}x_{2}}$ ${\displaystyle +}$ ${\displaystyle a_{13}x_{3}}$ ${\displaystyle +}$ ${\displaystyle \dots }$ ${\displaystyle =}$ ${\displaystyle \xi _{1},}$ ${\displaystyle a_{21}x_{1}}$ ${\displaystyle +}$ ${\displaystyle a_{22}x_{2}}$ ${\displaystyle +}$ ${\displaystyle a_{33}x_{3}}$ ${\displaystyle +}$ ${\displaystyle \dots }$ ${\displaystyle =}$ ${\displaystyle \xi _{2},}$ ${\displaystyle a_{31}x_{1}}$ ${\displaystyle +}$ ${\displaystyle a_{32}x_{2}}$ ${\displaystyle +}$ ${\displaystyle a_{33}x_{3}}$ ${\displaystyle +}$ ${\displaystyle \dots }$ ${\displaystyle =}$ ${\displaystyle \xi _{3},}$ we derive⁠ ${\displaystyle .}$ ${\displaystyle .}$ ${\displaystyle .}$ ${\displaystyle \dots }$ ${\displaystyle .}$

 ${\displaystyle \Delta x_{1}}$ ${\displaystyle =}$ ${\displaystyle {\mbox{A}}_{11}\xi _{1}}$ ${\displaystyle +}$ ${\displaystyle {\mbox{A}}_{21}\xi _{2}}$ ${\displaystyle +}$ ${\displaystyle {\mbox{A}}_{31}\xi _{3}}$ ${\displaystyle +}$ ${\displaystyle \dots ,}$ ${\displaystyle \Delta x_{2}}$ ${\displaystyle =}$ ${\displaystyle {\mbox{A}}_{12}\xi _{1}}$ ${\displaystyle +}$ ${\displaystyle {\mbox{A}}_{22}\xi _{2}}$ ${\displaystyle +}$ ${\displaystyle {\mbox{A}}_{32}\xi _{3}}$ ${\displaystyle +}$ ${\displaystyle \dots ,}$ ${\displaystyle \Delta x_{3}}$ ${\displaystyle =}$ ${\displaystyle {\mbox{A}}_{13}\xi _{1}}$ ${\displaystyle +}$ ${\displaystyle {\mbox{A}}_{23}\xi _{2}}$ ${\displaystyle +}$ ${\displaystyle {\mbox{A}}_{33}\xi _{3}}$ ${\displaystyle +}$ ${\displaystyle \dots ,}$ and thence⁠ ${\displaystyle .}$ ${\displaystyle .}$ ${\displaystyle .}$ ${\displaystyle .}$ ${\displaystyle \dots ,}$

 ${\displaystyle \Delta ^{n-1}x_{1}}$ ${\displaystyle =}$ ${\displaystyle {\mbox{B}}_{11}\Delta x_{1}}$ ${\displaystyle +}$ ${\displaystyle {\mbox{B}}_{12}\Delta x_{2}}$ ${\displaystyle +}$ ${\displaystyle {\mbox{B}}_{13}\Delta x_{3}}$ ${\displaystyle +}$ ${\displaystyle \dots ,}$ ${\displaystyle \Delta ^{n-1}x_{2}}$ ${\displaystyle =}$ ${\displaystyle {\mbox{B}}_{21}\Delta x_{1}}$ ${\displaystyle +}$ ${\displaystyle {\mbox{B}}_{22}\Delta x_{2}}$ ${\displaystyle +}$ ${\displaystyle {\mbox{B}}_{23}\Delta x_{3}}$ ${\displaystyle +}$ ${\displaystyle \dots ,}$ ${\displaystyle \Delta ^{n-1}x_{3}}$ ${\displaystyle =}$ ${\displaystyle {\mbox{B}}_{31}\Delta x_{1}}$ ${\displaystyle +}$ ${\displaystyle {\mbox{B}}_{32}\Delta x_{2}}$ ${\displaystyle +}$ ${\displaystyle {\mbox{B}}_{33}\Delta x_{3}}$ ${\displaystyle +}$ ${\displaystyle \dots ,}$ ${\displaystyle .}$ ${\displaystyle .}$ ${\displaystyle .}$ ${\displaystyle .}$ ${\displaystyle \dots ,}$

and comparison of the first and third systems yields

${\displaystyle {\mbox{B}}_{rs}=\Delta ^{n-2}a_{rs}}$.

In general it can be proved that any minor of order ${\displaystyle p}$ of the adjoint is equal to the complementary of the corresponding minor of the original multiplied by the (p – 1)th power of the original determinant.

Theorem.—The adjoint determinant is the (n – 1)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 ${\displaystyle a_{ik}=a_{ki}}$. Such determinants are called symmetrical. It is easy to see that the adjoint determinant is also symmetrical, viz. such that ${\displaystyle {\mbox{A}}_{ik}={\mbox{A}}_{ki}}$, for the determinant got by suppressing the i th row and kth column differs only by an interchange of rows and columns from that got by suppressing the kth row and i th column. If any symmetrical determinant vanish and be bordered as shown below

${\displaystyle {\begin{vmatrix}a_{11}&a_{12}&a_{13}&\Lambda _{1}\\a_{12}&a_{22}&a_{23}&\Lambda _{2}\\a_{13}&a_{23}&a_{33}&\Lambda _{3}\\\Lambda _{1}&\Lambda _{2}&\Lambda _{3}&.\end{vmatrix}}}$

it is a perfect square when considered as a function of ${\displaystyle \Lambda _{1},\Lambda _{2},\Lambda _{3}}$. For since ${\displaystyle {\mbox{A}}_{11}{\mbox{A}}_{22}-{\mbox{A}}_{12}^{3}=\Delta a_{33}}$, with similar relations, we have a number of relations similar to ${\displaystyle {\mbox{A}}_{11}{\mbox{A}}_{22}={\mbox{A}}_{12}^{2}}$, and either ${\displaystyle {\mbox{A}}_{rs}=+{\sqrt {(}}{\mbox{A}}_{rr}{\mbox{A}}_{ss})}$ or ${\displaystyle -{\sqrt {(}}{\mbox{A}}_{rr}{\mbox{A}}_{ss})}$ for all different values of ${\displaystyle r}$ and ${\displaystyle s}$. Now the determinant has the value

${\displaystyle -\{\lambda _{1}^{2}{\mbox{A}}_{11}+\lambda _{2}^{2}{\mbox{A}}_{22}+\lambda _{3}^{2}{\mbox{A}}_{33}+2\lambda _{2}\lambda _{3}{\mbox{A}}_{23}+2\lambda _{2}\lambda _{1}{\mbox{A}}_{31}+2\lambda _{1}\lambda _{2}{\mbox{A}}_{12}\}}$

${\displaystyle =-\Sigma \lambda _{r}^{2}{\mbox{A}}_{rr}-2\Sigma \lambda _{r}\lambda _{s}{\mbox{A}}_{rs}}$ in general, and hence by substitution

${\displaystyle \pm \{\lambda _{1}{\sqrt {\mbox{A}}}_{11}+\lambda _{2}{\sqrt {\mbox{A}}}_{22}+\dots +\lambda _{n}{\sqrt {\mbox{A}}}_{nn}\}^{2}.}$

A skew symmetric determinant has ${\displaystyle a_{rr}=0}$ and ${\displaystyle a_{rs}=-a_{sr}}$ for all values of ${\displaystyle r}$ and ${\displaystyle s}$. Such a determinant when of uneven degree vanishes, for if we multiply each row by ${\displaystyle -1}$ we multiply the determinant by ${\displaystyle (-1)^{n}=-1}$, 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 ${\displaystyle \Delta =-\Delta }$ or ${\displaystyle \Delta =0}$. 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

${\displaystyle {\mbox{A}}_{12}{\mbox{A}}_{34}-{\mbox{A}}_{13}{\mbox{A}}_{24}+{\mbox{A}}_{14}{\mbox{A}}_{23}}$.

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

${\displaystyle {\mbox{X}}=ax\ +by\ +cz,}$

${\displaystyle {\mbox{Y}}=a'x+b'y+c'z,}$

${\displaystyle {\mbox{Z}}=a''x+b''y+c''z,}$

where ${\displaystyle {\mbox{X}}^{2}+{\mbox{Y}}^{2}+{\mbox{Z}}^{2}=x^{2}+y^{2}+z^{2}}$. This relation implies six equations between the coefficients, so that only three of them are independent. Further we find

${\displaystyle x=a{\mbox{X}}+a'{\mbox{Y}}+a''{\mbox{Z}},}$

${\displaystyle y=b{\mbox{X}}+b'{\mbox{Y}}+b''{\mbox{Z}},}$

${\displaystyle z=c{\mbox{X}}+c'{\mbox{Y}}+c''{\mbox{Z}},}$

and the problem is to express the nine coefficients in terms of three independent quantities.

In general in space of ${\displaystyle n}$ dimensions we have ${\displaystyle n}$ substitutions similar to

${\displaystyle X_{1}=a_{11}x_{1}+A_{12}x_{2}+\dots +a_{1n}x_{n}}$,

and we have to express the ${\displaystyle n^{2}}$ coefficients in terms of ${\displaystyle {\tfrac {1}{2}}n(n-1)}$ independent quantities; which must be possible, because

 ${\displaystyle x_{1}}$ ${\displaystyle =}$ ${\displaystyle b_{11}\xi _{1}}$ ${\displaystyle +}$ ${\displaystyle b_{12}\xi _{2}}$ ${\displaystyle +}$ ${\displaystyle b_{13}\xi _{3}}$ ${\displaystyle +\dots ,}$ ${\displaystyle x_{2}}$ ${\displaystyle =}$ ${\displaystyle b_{21}\xi _{1}}$ ${\displaystyle +}$ ${\displaystyle b_{22}\xi _{2}}$ ${\displaystyle +}$ ${\displaystyle b_{23}\xi _{3}}$ ${\displaystyle +\dots ,}$ ${\displaystyle .}$ ${\displaystyle .}$ ${\displaystyle .}$ ${\displaystyle .}$ ${\displaystyle .}$ ${\displaystyle X_{1}}$ ${\displaystyle =}$ ${\displaystyle b_{11}\xi _{1}}$ ${\displaystyle +}$ ${\displaystyle b_{21}\xi _{2}}$ ${\displaystyle +}$ ${\displaystyle b_{31}\xi _{3}}$ ${\displaystyle +\dots ,}$ ${\displaystyle X_{1}}$ ${\displaystyle =}$ ${\displaystyle b_{12}\xi _{1}}$ ${\displaystyle +}$ ${\displaystyle b_{22}\xi _{2}}$ ${\displaystyle +}$ ${\displaystyle b_{32}\xi _{3}}$ ${\displaystyle +\dots ,}$ ${\displaystyle .}$ ${\displaystyle .}$ ${\displaystyle .}$ ${\displaystyle .}$ ${\displaystyle .}$

where ${\displaystyle b_{rr}=1}$ and ${\displaystyle b_{rs}=-b_{sr}}$ for all values of ${\displaystyle r}$ and ${\displaystyle s}$. There are then ${\displaystyle {\tfrac {1}{2}}n(n-1)}$ quantities ${\displaystyle b_{rs}}$. Let the determinant of the b’s be ${\displaystyle \Delta _{b}}$ and ${\displaystyle B_{rs}}$, the minor corresponding to ${\displaystyle b_{rs}}$. We can eliminate the quantities ${\displaystyle \xi _{1},\xi _{2},\dots \xi _{n}}$ and obtain ${\displaystyle n}$ relations

${\displaystyle \Delta _{b}{\text{X}}_{1}=(2{\text{B}}_{11}-\Delta _{b})x_{1}}$${\displaystyle +2{\text{B}}_{12}{\text{X}}_{2}+2{\text{B}}_{31}x_{3}+\dots ,}$

${\displaystyle \Delta _{b}{\text{X}}_{2}=}$${\displaystyle 2{\text{B}}_{12}x_{1}+(2{\text{B}}_{22}-\Delta _{b})x_{2}+2{\text{B}}_{32}x_{3}+\dots ,}$

${\displaystyle .}$${\displaystyle .}$${\displaystyle .}$${\displaystyle .}$${\displaystyle .}$${\displaystyle .}$${\displaystyle .}$

and from these another equivalent set

${\displaystyle \Delta _{b}x_{1}=(2{\text{B}}_{11}-\Delta _{b}){\text{X}}_{1}}$${\displaystyle +2{\text{B}}_{12}{\text{X}}_{2}+2{\text{B}}_{13}{\text{X}}_{3}+\dots ,}$

${\displaystyle \Delta _{b}x_{2}=}$${\displaystyle 2{\text{B}}_{21}{\text{X}}_{1}+(2{\text{B}}_{22}-\Delta _{b}){\text{X}}_{2}+2{\text{B}}_{23}{\text{X}}_{3}+\dots ,}$

${\displaystyle .}$${\displaystyle .}$${\displaystyle .}$${\displaystyle .}$${\displaystyle .}$${\displaystyle .}$${\displaystyle .}$

and now writing

${\displaystyle {\frac {2{\text{B}}_{ii}-\Delta _{b}}{\Delta _{b}}}=a_{ii},\qquad {\frac {2{\text{B}}_{ik}}{\Delta _{b}}}=a_{ik},}$

we have a transformation which is orthogonal, because ${\displaystyle \Sigma X^{2}=\Sigma x^{2}}$ and the elements ${\displaystyle a_{ii}}$, ${\displaystyle a_{ik}}$ are functions of the ${\displaystyle {\tfrac {1}{2}}n(n-1)}$ independent quantities ${\displaystyle b}$. We may therefore form an orthogonal transformation in association with every skew determinant which has its leading diagonal elements unity, for the ${\displaystyle {\tfrac {1}{2}}n(n-1)}$ quantities ${\displaystyle b}$ are clearly arbitrary.

For the second order we may take

${\displaystyle \Delta _{b}={\begin{vmatrix}1,&\lambda \\-\lambda ,&1\end{vmatrix}}=1+\Delta ^{2}}$,

and the adjoint determinant is the same; hence

${\displaystyle (1+\lambda ^{2})x_{1}=(1-\lambda ^{2}){\text{X}}_{1}+}$${\displaystyle 2\lambda {\text{X}}_{2},}$

${\displaystyle (1+\lambda ^{2})x_{2}}$${\displaystyle =-2\lambda {\text{X}}_{1}+(1-\lambda ^{2}){\text{X}}_{2}.}$

Similarly, for the order 3, we take

${\displaystyle \Delta _{b}={\begin{vmatrix}1&\nu &-\mu \\-\nu &1&\lambda \\\mu &-\lambda &1\end{vmatrix}}=1+\lambda ^{2}+\mu ^{2}+\nu ^{2},}$

${\displaystyle {\begin{vmatrix}1+\lambda ^{2}&\nu +\lambda \mu &-\mu +\lambda \nu \\-\nu +\lambda \mu &1+\mu ^{2}&\lambda +\mu \nu \\\mu +\lambda \nu &-\lambda +\mu \nu &1+\nu ^{2}\end{vmatrix}}}$,

 ${\displaystyle \Delta _{b}x_{1}=}$ ${\displaystyle (1+\lambda ^{2}-\mu ^{2}-\nu ^{2}){\text{X}}_{1}}$ ${\displaystyle +2(\nu +\lambda \mu ){\text{X}}_{2}}$ ${\displaystyle +2(-\mu +\lambda \nu ){\text{X}}_{3}}$ ${\displaystyle \Delta _{b}x_{2}=}$ ${\displaystyle 2(\lambda \mu -\nu ){\text{X}}_{1}}$ ${\displaystyle +(1+\mu ^{2}-\lambda ^{2}-\nu ^{2}){\text{X}}_{2}}$ ${\displaystyle +2(\mu \nu +\lambda ){\text{X}}_{3}}$ ${\displaystyle \Delta _{b}x_{3}=}$ ${\displaystyle 2(\lambda \nu +\mu ){\text{X}}_{1}}$ ${\displaystyle +2(\mu \nu -\lambda ){\text{X}}_{2}}$ ${\displaystyle +(1+\nu ^{2}-\lambda ^{2}-\mu ^{2}){\text{X}}_{3}}$.
Functional determinants were first investigated by Jacobi in a work De Determinantibus Functionalibus. Suppose ${\displaystyle n}$ dependent variables ${\displaystyle y_{1},y_{2},\dots y_{n}}$, each of which is a function of ${\displaystyle n}$ independent variables ${\displaystyle x_{1},x_{2},\dots x_{n}}$, so that ${\displaystyle y_{s}=f_{s}(x_{1},x_{2},\dots x_{n})}$. From the differential coefficients of the y’s with regard to the x’s we form the functional determinant

${\displaystyle {\text{R}}={\begin{vmatrix}{\frac {\partial y_{1}}{\partial x_{1}}}&{\frac {\partial y_{1}}{\partial x_{2}}}&\dots &{\frac {\partial y_{1}}{\partial x_{n}}}\\{\frac {\partial y_{2}}{\partial x_{1}}}&{\frac {\partial y_{2}}{\partial x_{2}}}&\dots &{\frac {\partial y_{2}}{\partial x_{n}}}\\.&.&\dots &.\\{\frac {\partial y_{n}}{\partial x_{1}}}&{\frac {\partial y_{n}}{\partial x_{2}}}&\dots &{\frac {\partial y_{n}}{\partial x_{n}}}\\\end{vmatrix}}{\begin{matrix}={\binom {y_{1},~y_{2},\dots y_{n}}{x_{1},~x_{2},\dots x_{n}}}\\{\text{for brevity.}}\end{matrix}}}$

If we have new variables z such that zs=φs(y1, y2,...yn), we have also zs = ψs(x1, x2,...xn), and we may consider the three determinants

(y1, y2,...yn
x1, x2,...xn
), (z1, z2,...zn
y1, y2,...yn
), (z1, z2,...zn
x1, x2,...xn
)

Forming the product of the first two by the product theorem, we obtain for the element in the ith row and kth column

ziy1 y1xk+ziy2 y2xk+...+ziyn ynxk

which is zixk, the partial differential coefficient of zi, with regard to xk . Hence the product theorem

(z1, z2,...zn
y1, y2,...yn
), (y1, y2,...yn
x1, x2,...xn
) = (z1, z2,...zn
x1, x2,...xn
);
and as a particular case
(y1, y2,...yn
x1, x2,...xn
) (x1, x2,...xn
y1, y2,...yn
) = 1.

Theorem.—If the functions y1, y2,...yn be not independent of one another the functional determinant vanishes, and conversely if the determinant vanishes, y1, y2,...yn 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

ƒ1 = a11x1+ a12x2+ ... a1nxn=0,
ƒ2 = a21x1+ a22x2+ ... a2nxn=0,
.......
ƒn = an1x1+ an2x2+ ... annxn=0.

Denote by Δ the determinant (a11a22...ann).

Multiplying the equations by the minors A1μ, A2μ,...Anμ respectively, and adding, we obtain

xμ(a1μA1μ+a2μA2μ+...+anμAnμ) = xμΔ = 0,

since from results already given the remaining coefficients of x1, x2,...xμ–1, xμ+1,...xn vanish identically.

Hence if Δ does not vanish x1 = x1 = ... =xn = 0 is the only solution; but if Δ 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

A1μƒ1 + A2μƒ2+...+Anμƒn = 0,

and assuming that the minors do not all vanish the satisfaction of n–1 of the equations implies the satisfaction of the nth.

Consider then the system of n–1 equations

a21x1+ a22x2 +...+ a2nxn = 0
a31x1+ a32x2 +...+ a3nxn = 0
......
an1x1+ an2x2 +...+ annxn = 0,
which becomes on writing xsxn = ys,
a21y1+ a22y2 +...+ a2,n−1yn−1 +a2n = 0
a31y1+ a32y2 +...+ a3,n−1yn−1 +a3n = 0
.......
an1y1+ an2y2 +...+ an,n−1yn−1 +ann = 0.
We can solve these, assuming them independent, for the n−1 ratios y1, y2,...yn−1.
Now
a21A11 + a22A12+...+a2nA1n = 0
a31A11 + a32A12+...+a3nA1n = 0
.......
an1A11 + an2A12+...+annA1n = 0
and therefore, by comparison with the given equations, xi = ρA1i, where ρ is an arbitrary factor which remains constant as i varies.

Hence yi = A1iA1n where A li and A1n, are minors of the complete determinant
(a11a22...ann).

 a21 a22 ...a2,i–1 a2,i+1... a2n a31 a32 ...a3,i–1 a3,i+1 ...a3n ⁠.⁠.⁠...⁠.⁠.⁠...⁠. ⁠ ∴ yi = (−)i+n an1 an2 ...an,i–1 an,i+1 ...a2nn .mw-parser-output .wst-bar{text-decoration:line-through}.mw-parser-output .wst-bar-inner{color:transparent}————————————,
 ⁠ a21 a22 ...a2,n⁠–1 a31 a22 ...a2,n⁠–1 ⁠.⁠.⁠...⁠. an1 an2 ...an,n⁠–1

or, in words, yi is the quotient of the determinant obtained by erasing the i th column by that obtained by erasing the nth column, multiplied by (–1)i+n. For further information concerning the compatibility and independence of a system of linear equations, see Gordon, Vorlesungen über Invariantentheorie, Bd. 1, § 8.

Resultants.—When we are given k homogeneous equations in k variables or k non-homogeneous equations in k − 1 variables, the equations being independent, it is always possible to derive from them a single equation R = 0, 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 = 0 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.

ƒ(x) = ƒ = a0xma1xm–1 + a2xm–2 – ... = 0,
ƒ(φ) = φ = b0xnb1xn–1 + b2xn–2 – ... = 0,
If α1, α2, ...αm be the roots of ƒ=0, β1, β2, ...βn the roots of φ=0, the condition that some root of 0 =o may qq cause f to vanish is clearly
Rƒ,φ = ƒ (β1)ƒ(β2)...ƒ(β2) = 0;
so that Rƒ,φ is the resultant of ƒ and φ, and expressed as a function of the roots, it is of degree m in each root β, and of degree n in each root α, and also a symmetric function alike of the roots α and of the roots β; hence, expressed in terms of the coefficients, it is homogeneous and of degree n in the coefficients of ƒ, and homogeneous and of degree m in the coefficients of φ
Ex. gr.
ƒ = a0x² − a1x+a2 =0, φ=b0x² − b1x+b2.
We have to multiply a0β2
1
a1β1+a2 by a0β2
2
a1β2+a2 and we obtain
a2
0
β2
1
β2
2
a0a1(β2
1
β2 + β1β2
2
) + a0a2(β2
1
β2
1
+ β1β2
2
) + a2
1
0
β1β2a1a2(β1 + β2) + a2
2
,
where
β1 + β2 = b1b0,β1 β2 = b2b0, β1 β2 = b2
1
– 2b0b2
b2
0
,
and clearing of fractions
Rƒ,φ = (a0b2a2b0)² + (a1b0a0b1)(a1b2a2b1).

We may equally express the result as
φ(α(1)φ(α2)...φ(αm) = 0,
II
s,t
(αsβt = 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 mnin 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 +...+ a1pxp = 0,
a21x1 + a22x2 +...+ a2pxp = 0,
......
ap1x1 + ap2x2 +...+ appxp = 0,
be the system the condition is, in determinant form,
(a11a22...app) = 0;
n fact the determinant is the resultant of the equations.

Now, suppose ƒ and φ to have a common factor xγ,
ƒ(x) =ƒ1(x)(xγ); φ(x) = φ1(x)(xγ),
ƒ1 and φ1 being of degrees m – 1 and n – 1 respectively; we have the identity φ1ƒ(x) = ƒ1(x)φ(x) of degree m + n – 1.

Assuming then φ1 to have the coefficients B1, B2,...Bn
and ƒ1the coefficients A1, A2,...Am,
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, B2,...Bn, A1, A2,...Am. Forming the resultant of these equations we evidently obtain the resultant of ƒ and φ.
Thus to obtain the resultant of
ƒ=a0x3 + a1x2 + a2x+ a3, , φ = b0x2 + b1x+ b2
we assume the identity
(B0x + B1)(a0x3 + a1x2 + a2x+ a3) = (A0x2 + A1x+ A2)(b0x2 + b1x+ b2),
and derive the linear equations

 B0a0 −A0b0 =0, B0a1 +B1a0 −A0b1 −A1b0 =0, B0a2 +B1a1 −A0b2 −A1b1 −A2b0 =0, B0a3 +B1a2 −A1b2 −A2b1 =0, ⁠B1a3 −A2b2 =0,
and by elimination we obtain the resultant

${\displaystyle {\begin{vmatrix}a_{0}&0&b_{0}&0&0\\a_{1}&a_{0}&b_{1}&b_{0}&0\\a_{2}&a_{1}&b_{2}&b_{1}&b_{0}\\a_{3}&a_{2}&0&b_{2}&b_{1}\\0&a_{3}&0&0&b_{2}\\\end{vmatrix}}~{\begin{matrix}{\text{a numerical factor}}\\{\text{being disregarded.}}\end{matrix}}}$

This is Euler’s method. Sylvester’s leads to the same expression, but in a simpler manner.

He forms n equations from ƒ by separate multiplication by xn–1, xn–2,...x, 1, in succession, and similarly treats φ with m multipliers xm –1, xm –2,...x, 1. From these m + n equations he eliminates the m + n powers xm+n –1, xm+n–2, x,.. 1, treating them as independent unknowns. Taking the same example as before the process leads to the system of equations

 a0x4+ a1x3+ a2x2+ a3x =0, a0x3+ a1x2+ a2x+ a3 =0, b0x4+ b1x3+ b2x2 =0, b0x3+ b1x2+ b2x =0, b0x2+ b1x+ b2 =0,

whence by elimination the resultant

 a0 a1 a2 a3 0 0 a0 a1 a2 a3 b0 b1 b2 0 0 0 b0 b1 b2 0 0 0 b9 b1 b2

which reads by columns as the former determinant reads by rows, and is therefore identical with the former. E. Bézout’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
ƒ(x)φ(x′) − ƒ(x′)φ(x) = 0,
which can be satisfied when ƒ and φ possess a common factor. He first divides by the factor xx′, reducing it to the degree m − 1 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 x0, x, x2,... xm−1 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
(a0x3+a1x2+a2x +a3) (b0x2+b1x′+b2) − (a0x3+a1x2+a2x′ +a3) (b0x2+b1x+b2) = 0;
after division by xx′ the three equations are formed

 a0b0x2+a0b1x+a0b2 = 0, a0b1x2+(a0b2+a1b1−a0b2)x+a1b2−a3b0 = 0, a0b2x2+(a1b2−a3b0)x+a2b2−a3b1 = 0

and thence the resultant

 a0b0 a0b1 a0b2 a0b1 a0b2+a1b1−a0b2 a1b2−a3b0 a0b2 a1b2−a3b0 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 (u1 v2 w3), and by Euler’s theorem of homogeneous functions
xu1 + yu2 + zu3 = mu
xv1 + yv2 + zv3 = nv
xw1 + yw2 + zw3 = pw;
denoting now the reciprocal determinant by (U1 V2 W3) we obtain Jx = muU1 + nvV1 + pwW1; Jy=..., Jz=..., and it appears that the vanishing of u, v, and w implies the vanishing of J. Further, if m = n = p, we obtain by differentiation
J + x∂Jx =m (u∂U1x. + v∂V1x + u∂W1x + u1U1 v1V1 w1W1).
or
x∂Jx =m – 1)J + m (u∂U1x. + v∂V1x + u∂W1x).

Hence the system of values also causes ∂Jx to vanish in this case; and by symmetry ∂Jy and ∂Jz also vanish.

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
axm + (by + cz) xm–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, combined 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 x², y², z², yz, zx, xy from the six equations
u = v = w = ∂Jx = ∂Jy = ∂Jz = 0; if we apply the same process to these 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–1, and thus contains the coefficients of each form to the degree (m–1)k–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–1)k–1 (Salmon, l.c. p. 100).

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, ... Xk – 1, the coefficients being any polynomials, it is clear that the k differentials have, in common, the system of roots derived from X1 = X2 = ... = Xk – 1 = 0, and have in consequence a vanishing resultant. This implies the vanishing of the discriminant of the original form.

Expression in Terms of Roots.—Since x∂ƒx+∂ƒy = mƒ, if we take any root x1, y1, of ∂ƒx, and substitute in mf we must obtain, y1(∂ƒy)
xx1

yy1
; hence the resultant of ∂ƒx and ƒ is, disregarding numerical factors, y1y2...yn–1 × discriminant of ƒ = a0 × disct. of ƒ.

Now
ƒ = (xy1x1y)(xy2x2y) ... (xymxmy),
∂ƒx =Σ1 y1(xymxmy),
and substituting in the latter any root of ƒ and forming the product, we find the resultant of ƒ and ∂ƒx, viz.

y1y2...ym(x1y2x2y1)2(x1y3x3y1)2...(xrysxsyr)2...

and, dividing by y1y2...ym, the discriminant of ƒ 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
II(αrαs)2II(βrβs)2{II(αrβs}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 ${\displaystyle n}$ quantities ${\displaystyle a_{1},a_{2},a_{3},\dots a_{n}}$.

Every rational integral function of these quantities, which does not alter its value however the ${\displaystyle n}$ suffixes ${\displaystyle 1,2,3,\dots n}$ be permuted, is a rational integral symmetric function of the quantities. If we write ${\displaystyle (1+a_{1}x)(1+a_{2}x)\dots (1+a_{n}x)=1+a_{1}x+a_{2}x^{2}+\dots +a_{n}x^{n}}$, ${\displaystyle a_{1},a_{2},\dots a_{n}}$ are called the elementary symmetric functions.

${\displaystyle a_{1}=a_{1}+a_{2}+\dots +a_{n}=\Sigma a_{1}}$
${\displaystyle a_{2}=a_{1}a_{2}+a_{2}a_{3}+a_{2}a_{3}+\dots =\Sigma a_{1}a_{2}}$

${\displaystyle a_{n}=a_{1}a_{2}a_{3}\dots a_{n}}$

The general monomial symmetric function is

${\displaystyle \Sigma a_{1}^{p_{1}}a_{2}^{p_{2}}a_{3}^{p_{3}}\dots a_{n}^{p_{n}}}$,

the summation being for all permutations of the indices which result in different terms. The function is written

${\displaystyle (p_{1}p_{2}p_{3}\dots p_{n})}$

for brevity, and repetitions of numbers in the bracket are indicated by exponents, so that ${\displaystyle (p_{1}p_{1}p_{2})}$ is written ${\displaystyle (p_{1}^{2}p_{2}).}$ 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

${\displaystyle (1),(1^{2}),(1^{3}),\dots (1^{n})}$,

are all of the first degree, and are of weights ${\displaystyle 1,2,3,\dots n}$ respectively.

Remark.—In this notation ${\displaystyle (0)=\Sigma a_{1}^{0}={\tbinom {n}{1}}}$; ${\displaystyle (0^{2})=\Sigma a_{1}^{0}a_{2}^{0}={\tbinom {n}{2}}}$; ... ${\displaystyle (0^{s})={\tbinom {n}{s}}}$, &c. The binomial coefficients appear, in fact, as symmetric functions, and this is frequently of importance.

The order of the numbers in the bracket ${\displaystyle (p_{1}p_{2}\dots p_{n})}$ 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 ${\displaystyle w}$, 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 ${\displaystyle h_{w}}$; it is connected with the elementary functions by the formula

${\displaystyle {\frac {1}{1-a_{1}x+a_{2}x^{2}-a_{3}x^{3}+\dots }}=1+h_{1}x+h_{2}x^{2}+h_{3}x^{3}+\dots }$,

which remains true when the symbols ${\displaystyle a}$ and ${\displaystyle h}$ are interchanged, as is at once evident by writing ${\displaystyle -x}$ for ${\displaystyle x}$. This proves, also, that in any formula connecting ${\displaystyle a_{1},a_{2},a_{3},\dots }$ with ${\displaystyle h_{1},h_{2},h_{3},\dots }$ the symbols ${\displaystyle a}$ and ${\displaystyle h}$ may be interchanged.

Ex. gr, from ${\displaystyle h_{2}=a_{1}^{2}-a_{2}}$ we derive ${\displaystyle a_{2}=h_{1}^{2}-h_{2}}$.

The function ${\displaystyle \Sigma a_{1}^{p_{1}}a_{2}^{p_{2}}\dots a_{n}^{p_{n}}}$ being as above denoted by a partition of the weight, viz. ${\displaystyle (p_{1}p_{2}\dots p_{n})}$, it is necessary to bring under view other functions associated with the same series of numbers: such, for example, as

${\displaystyle \Sigma a_{1}^{p_{1}}a_{2}^{p_{3}}\Sigma a_{1}^{p_{2}}a_{2}^{p_{4}}\dots a_{n-2}^{p_{n-2}}=(p_{1}p_{3})(p_{2}p_{4}\dots p_{n-2})}$.

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 ${\displaystyle w_{1},w_{2},w_{3},\dots }$ be enclosed in a bracket we obtain a partition of the weight ${\displaystyle w}$ 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, ${\displaystyle (p_{1}p_{1}p_{1}p_{2}p_{2}p_{3})=(p_{1}^{3}p_{2}^{2}p_{3})}$, can be separated in the manner ${\displaystyle (p_{1}p_{2})(p_{1}p_{2})(p_{1}p_{3})=(p_{1}p_{2})^{2}(p_{1}p_{3})}$, and we may take the general form of a partition to be ${\displaystyle (p_{1}^{\pi _{1}}p_{2}^{\pi _{2}}p_{3}^{\pi _{3}}\dots )}$ and that of a separation ${\displaystyle ({\text{J}}_{1})^{j_{1}}({\text{J}}_{2})^{j_{2}}({\text{J}}_{3})^{j_{3}}\dots }$ when ${\displaystyle {\text{J}}_{1},{\text{J}}_{2},{\text{J}}_{3}\dots }$ denote the distinct separates involved.

Theorem.— The function symbolized by ${\displaystyle (n)}$, viz. the sum of the nth powers of the quantities, is expressible in terms of functions which are symbolized by separations of any partition ${\displaystyle (n_{1}^{\nu _{1}}n_{2}^{\nu _{2}}n_{3}^{\nu _{3}}\ldots )}$ of the number ${\displaystyle n}$. The expression is—

${\displaystyle (-)^{\nu _{1}+\nu _{2}+\nu _{3}+\ldots }{\frac {(\nu _{1}+\nu _{2}+\nu _{3}+\dots -1){\text{!}}}{\nu _{1}{\text{!}}\nu _{2}{\text{!}}\nu _{3}{\text{!}}\ldots }}(n)}$
${\displaystyle =\sum (-)^{j_{1}+j_{2}+j_{3}+\ldots }{\frac {(j_{1}+j_{2}+j_{3}+\dots -1){\text{!}}}{j_{1}{\text{!}}j_{2}{\text{!}}j_{3}{\text{!}}\ldots }}({\text{J}}_{1})^{j_{1}}({\text{J}}_{2})^{j_{2}}({\text{J}}_{3})^{j_{3}}\dots }$,

${\displaystyle ({\text{J}}_{1})^{j_{1}}({\text{J}}_{2})^{j_{2}}({\text{J}}_{3})^{j_{3}}\dots }$ being a separation of ${\displaystyle (n_{1}^{\nu _{1}}n_{2}^{\nu _{2}}n_{3}^{\nu _{3}}\ldots )}$ and the summation being in regard to all such separations. For the particular case ${\displaystyle (n_{1}^{\nu _{1}}n_{2}^{\nu _{2}}n_{3}^{\nu _{3}}\ldots )=(1^{n})}$

${\displaystyle (-)^{n}{\frac {1}{n}}(n)=\sum (-)^{j_{1}+j_{2}+j_{3}+\ldots }{\frac {(j_{1}+j_{2}+j_{3}+\dots -1){\text{!}}}{j_{1}{\text{!}}j_{2}{\text{!}}j_{3}{\text{!}}\ldots }}(1_{1})^{j_{1}}(1_{2})^{j_{2}}(1_{3})^{j_{3}}\dots }$

To establish this write—

${\displaystyle 1+\mu {\text{X}}_{1}+\mu ^{2}{\text{X}}_{2}+\mu ^{3}{\text{X}}_{3}+\dots ={\underset {a}{\text{II}}}(1+\mu a_{1}x_{1}+\mu ^{2}a_{1}^{2}x_{2}+\mu ^{3}a_{1}^{3}x_{3}+\dots )}$,

the product on the right involving a factor for each of the quantities ${\displaystyle a_{1},a_{2},a_{3}\dots }$, and ${\displaystyle \mu }$ being arbitrary.

Multiplying out the right-hand side and comparing coefficients

${\displaystyle {\text{X}}_{1}=(1)x_{1}}$,
${\displaystyle {\text{X}}_{2}=(2)x_{2}+(1^{2})x_{1}^{2}}$,
${\displaystyle {\text{X}}_{3}=(3)x_{3}+(21)x_{2}x_{1}+(1^{3})x_{1}^{3}}$,
${\displaystyle {\text{X}}_{4}=(4)x_{4}+(31)x_{3}x_{1}+(2^{2})x_{2}^{2}+(21^{2})x_{2}x_{1}^{2}+(1^{4})x_{1}^{4}}$,

${\displaystyle {\text{X}}_{m}=\Sigma (m_{1}^{\mu _{1}}m_{2}^{\mu _{2}}m_{3}^{\mu _{3}}\dots )x_{m_{1}}^{\mu _{1}}x_{m_{2}}^{\mu _{2}}x_{m_{3}}^{\mu _{3}}\dots }$,

the summation being for all partitions of ${\displaystyle m}$.

Auxiliary Theorem.—The coefficient of ${\displaystyle x_{l_{1}}^{\lambda _{1}}x_{l_{2}}^{\lambda _{2}}x_{l_{3}}^{\lambda _{3}}\dots }$ in the product ${\displaystyle {\frac {{\text{X}}_{m_{1}}^{\mu _{1}}{\text{X}}_{m_{2}}^{\mu _{2}}{\text{X}}_{m_{3}}^{\mu _{3}}\dots }{\mu _{1}{\text{!}}\mu _{2}{\text{!}}\mu _{3}{\text{!}}\dots }}}$ is ${\displaystyle \sum {\frac {({\text{J}}_{1})^{j_{1}}({\text{J}}_{2})^{j_{2}}({\text{J}}_{3})^{j_{3}}\dots }{j_{1}{\text{!}}j_{2}{\text{!}}j_{3}{\text{!}}\dots }}}$ where ${\displaystyle ({\text{J}}_{1})^{j_{1}}({\text{J}}_{2})^{j_{2}}({\text{J}}_{3})^{j_{3}}\dots }$ is a separation of ${\displaystyle (l_{1}^{\lambda _{1}}l_{2}^{\lambda _{2}}l_{3}^{\lambda _{3}}\dots )}$ of specification ${\displaystyle (m_{1}^{\mu _{1}}m_{2}^{\mu _{2}}m_{3}^{\mu _{3}}\dots )}$, and the sum is for all such separations.

To establish this observe the result.

${\displaystyle {\frac {1}{p{\text{!}}}}{\text{X}}_{3}^{p}=\sum {\frac {(3)^{\pi _{1}}(21)^{\pi _{2}}(1^{3})^{\pi _{3}}}{\pi _{1}{\text{!}}\pi _{2}{\text{!}}\pi _{3}{\text{!}}}}x_{3}^{\pi _{1}}x_{2}^{\pi _{2}}x_{1}^{\pi _{2}+3\pi _{3}}}$

and remark that ${\displaystyle (3)^{\pi _{1}}(21)^{\pi _{2}}(1^{3})^{\pi _{3}}}$ is a separation of ${\displaystyle (3^{\pi _{1}}2^{\pi _{2}}1^{\pi _{2}+3\pi _{3}})}$ of specification ${\displaystyle (3^{p})}$. A similar remark may be made in respect of

${\displaystyle {\frac {1}{\mu _{1}{\text{!}}}}{\text{X}}_{m_{1}}^{\mu _{1}},{\frac {2}{\mu _{2}{\text{!}}}}{\text{X}}_{m_{2}}^{\mu _{2}},{\frac {3}{\mu _{3}{\text{!}}}}{\text{X}}_{m_{3}}^{\mu _{3}},\dots }$,

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

Now

${\displaystyle \log(1+\mu {\text{X}}_{1}+\mu ^{2}{\text{X}}_{2}+\mu ^{3}{\text{X}}_{3}+\dots )}$
${\displaystyle ={\underset {a}{\Sigma }}\log(1+\mu a_{1}x_{1}+\mu ^{2}a_{1}^{2}x_{2}+\mu ^{3}a_{1}^{3}x_{3}+\dots )}$

whence, expanding by the exponential and multinomial theorems, a comparison of the coefficients of ${\displaystyle \mu ^{n}}$ gives

${\displaystyle (n)\sum (-)^{\nu _{1}+\nu _{2}+\nu _{3}+\ldots -1}{\frac {(\nu _{1}+\nu _{2}+\nu _{3}+\ldots -1){\text{!}}}{\nu _{1}{\text{!}}\nu _{2}{\text{!}}\nu _{3}{\text{!}}\ldots }}x_{n_{1}}^{\nu _{1}}x_{n_{2}}^{\nu _{2}}x_{n_{3}}^{\nu _{3}}\dots }$
${\displaystyle =\sum (-)^{\nu _{1}+\nu _{2}+\nu _{3}+\ldots -1}{\frac {(\nu _{1}+\nu _{2}+\nu _{3}+\ldots -1){\text{!}}}{\nu _{1}{\text{!}}\nu _{2}{\text{!}}\nu _{3}{\text{!}}\ldots }}{\text{X}}_{n_{1}}^{\nu _{1}}{\text{X}}_{n_{2}}^{\nu _{2}}{\text{X}}_{n_{3}}^{\nu _{3}}\dots }$

and, by the auxiliary theorem, any term ${\displaystyle {\text{X}}_{m_{1}}^{\mu _{1}}{\text{X}}_{m_{2}}^{\mu _{2}}{\text{X}}_{m_{3}}^{\mu _{3}}\dots }$ on the right-hand side is such that the coefficient of ${\displaystyle x_{n_{1}}^{\nu _{1}}x_{n_{2}}^{\nu _{2}}x_{n_{3}}^{\nu _{3}}\dots }$ in ${\displaystyle {\frac {1}{\mu _{1}{\text{!}}\mu _{2}{\text{!}}\mu _{3}{\text{!}}\ldots }}{\text{X}}_{m_{1}}^{\mu _{1}}{\text{X}}_{m_{2}}^{\mu _{2}}{\text{X}}_{m_{3}}^{\mu _{3}}\dots }$ is

${\displaystyle \sum {\frac {({\text{J}}_{1})^{j_{1}}({\text{J}}_{2})^{j_{2}}({\text{J}}_{3})^{j_{3}}\dots }{j_{1}{\text{!}}j_{2}{\text{!}}j_{3}{\text{!}}\ldots }}}$,

where since ${\displaystyle (m_{1}^{\mu _{1}}m_{2}^{\mu _{2}}m_{3}^{\mu _{3}}\dots )}$ is the specification of ${\displaystyle ({\text{J}}_{1})^{j_{1}}({\text{J}}_{2})^{j_{2}}({\text{J}}_{3})^{j_{3}}\dots }$, ${\displaystyle \mu _{1}+\mu _{2}+\mu _{3}+\dots =j_{1}+j_{2}+j_{3}+\dots }$. Comparison of the coefficients of ${\displaystyle x_{n_{1}}^{\nu _{1}}x_{n_{2}}^{\nu _{2}}x_{n_{3}}^{\nu _{3}}\dots }$ therefore yields the result

${\displaystyle (-)^{\nu _{1}+\nu _{2}+\nu _{3}+\ldots }{\frac {(\nu _{1}+\nu _{2}+\nu _{3}+\ldots -1){\text{!}}}{\nu _{1}{\text{!}}\nu _{2}{\text{!}}\nu _{3}{\text{!}}\ldots }}(n)}$
${\displaystyle =\sum (-)^{j_{1}+j_{2}+j_{3}+\ldots }{\frac {(j_{1}+j_{2}+j_{3}+\ldots -1){\text{!}}}{j_{1}{\text{!}}j_{2}{\text{!}}j_{3}{\text{!}}\ldots }}({\text{J}}_{1})^{j_{1}}({\text{J}}_{2})^{j_{2}}({\text{J}}_{3})^{j_{3}}\dots }$,

for the expression of ${\displaystyle \Sigma a^{n}}$ in terms of products of symmetric functions symbolized by separations of ${\displaystyle (n_{1}^{\nu _{1}}n_{2}^{\nu _{2}}n_{3}^{\nu _{3}}\dots )}$.

Let ${\displaystyle (n)_{a},(n)_{x},(n)_{\text{x}}}$ denote the sums of the nth powers of quantities whose elementary symmetric functions are ${\displaystyle a_{1},a_{2},a_{3},\dots }$; ${\displaystyle x_{1},x_{2},x_{3},\dots }$; ${\displaystyle {\text{X}}_{1},{\text{X}}_{2},{\text{X}}_{3},\dots }$ respectively: then the result arrived at above from the logarithmic expansion may be written

${\displaystyle (n)_{a}(n)_{x}=(n){\text{x}}}$,

exhibiting ${\displaystyle (n)_{\text{x}}}$ as an invariant of the transformation given by the expressions of ${\displaystyle {\text{X}}_{1},{\text{X}}_{2},{\text{X}}_{3},\dots }$ in terms of ${\displaystyle x_{1},x_{2},x_{3},\dots }$.

The inverse question is the expression of any monomial symmetric function by means of the power functions ${\displaystyle (r)=s_{r}}$.

Theorem of Reciprocity.—If

${\displaystyle {\text{X}}_{m_{1}}^{\mu _{1}}{\text{X}}_{m_{2}}^{\mu _{2}}{\text{X}}_{m_{3}}^{\mu _{3}}\dots =\dots +\theta (s_{1}^{\sigma _{1}}s_{2}^{\sigma _{2}}s_{3}^{\sigma _{3}}\dots )x_{l_{1}}^{\lambda _{1}}x_{l_{2}}^{\lambda _{2}}x_{l_{3}}^{\lambda _{3}}\dots +\dots }$,

where ${\displaystyle \theta }$ is a numerical coefficient, then also

${\displaystyle {\text{X}}_{s_{1}}^{\sigma _{1}}{\text{X}}_{s_{2}}^{\sigma _{2}}{\text{X}}_{s_{3}}^{\sigma _{3}}\dots =\dots +\theta (m_{1}^{\mu _{1}}m_{2}^{\mu _{2}}m_{3}^{\mu _{3}}\dots )x_{l_{1}}^{\lambda _{1}}x_{l_{2}}^{\lambda _{2}}x_{l_{3}}^{\lambda _{3}}\dots +\dots }$.

We have found above that the coefficient of ${\displaystyle (x_{l_{1}}^{\lambda _{1}}x_{l_{2}}^{\lambda _{2}}x_{l_{3}}^{\lambda _{3}}\dots )}$ in the product ${\displaystyle {\text{X}}_{m_{1}}^{\mu _{1}}{\text{X}}_{m_{2}}^{\mu _{2}}{\text{X}}_{m_{3}}^{\mu _{3}}\dots }$ is

${\displaystyle {\mu _{1}{\text{!}}\mu _{2}{\text{!}}\mu _{3}{\text{!}}\ldots }\sum {\frac {({\text{J}}_{1})^{j_{1}}({\text{J}}_{2})^{j_{2}}({\text{J}}_{3})^{j_{3}}\dots }{j_{1}{\text{!}}j_{2}{\text{!}}j_{3}{\text{!}}\ldots }}}$,

the sum being for all separations of ${\displaystyle l_{1}^{\lambda _{1}}l_{2}^{\lambda _{2}}l^{\lambda _{3}}\dots )}$ which have the specification ${\displaystyle (m_{1}^{\mu _{1}}m_{2}^{\mu _{2}}m^{\mu _{3}}\dots )}$. We can multiply out this expression so as to obtain a series of monomials of the form ${\displaystyle \theta (s_{1}^{\sigma _{1}}s_{2}^{\sigma _{2}}s_{3}^{\sigma _{3}}\dots )}$. It can be shown that the number ${\displaystyle \theta }$ enumerates distributions of a certain nature defined by the partitions ${\displaystyle (m_{1}^{\mu _{1}}m_{2}^{\mu _{2}}\dots )}$, ${\displaystyle (s_{1}^{\sigma _{1}}s_{2}^{\sigma _{2}}\dots )}$, (lλ1
1
lλ2
2
...) and it is seen intuitively that the number θ remains unaltered when the first two of these partitions are interchanged (see Combinatorial Analysis). Hence the theorem is established.

Putting x1= 1 and x2 = x3 = x4 = ... = 0, we find a particular law of reciprocity given by Cayley and Betti,

(1m1)μ1(1m2)μ2(1m3)μ3... = ... + θ(sσ1
1
sσ2
2
sσ3
3
...) + ...,
(1s1)σ1(1s2)σ2(1s3)σ3... = ... + θ(mμ1
1
mμ2
2
mμ3
3
...) + ...;

and another by putting x1 = x2 = x3 = ... = 1, for then Xm becomes hm, and we have
hμ1
m1
hμ2
m2
hμ3
m3
... = ... + θ′(sσ1
1
sσ2
2
sσ3
3
...) + ...,
hσ1
s1
hσ2
s2
hσ3
s3
... = ... + θ′(mμ1
1
mμ2
2
mμ3
3
...) + ...;

Theorem of Expressibility.—"If a symmetric function be symboilized by (λμν...) and (λ1λ2λ3...), (μ1μ2μ3...), (ν1ν2ν3...)... be any partitions of λ, μ, ν... respectively, the function (λμν...) is expressible by means of functions symbolized by separation of
(λ1λ2λ3...μ1μ2μ3...ν1ν2ν3...)."
For, writing as before,

 Xμ1m1 Xμ2m2 Xμ3m3 ... = ΣΣθ(sσ11sσ22sσ33...) xλ1l1 xλ2l2 xλ3l3..., = ΣP xλ1l1 xλ2l2 xλ3l3...,

P is a linear function of separations of (lλ1
1
lλ2
2
lλ3
3
...) of specification (mμ1
1
mμ2
2
mμ3
3
...), and if Xσ1
s1
Xσ2
s2
Xσ3
s3
... = ΣP′ xλ1
l1
xλ2
l2
xλ3
l3
..., P′ is a linear function of separations (lλ1
1
lλ2
2
lλ3
3
...) of specification (sσ1
1
sσ2
2
sσ3
3
...). Suppose the separations of (lλ1
1
lλ2
2
lλ3
3
...) to involve k different specifications and form the k identities
Xμ1
m1
Xμ2
m2
Xμ3
m3
... = Σ P(s) xλ1
l1
xλ2
l2
xλ3
l3
...(s = 1, 2, ...k),
where (mμ1s
1s
mμ2s
2s
mμ3s
3s
....) is one of the k specifications.

The law of reciprocity shows that
P(s) = ${\displaystyle \sum _{l=1}^{l=k}}$ θst(mμ1l
1l
mμ2l
2l
mμ3l
3l
...)
viz.: a linear function of symmetric functions symbolized by the k specifications; and that θst = θts. A table may be formed expressing the k expressions P(1), P(3),...P(k) as linear functions of the k expressions (mμ1s
1s
mμ2s
2s
mμ3s
3s
....), s = 1, 2, ...k, and the numbers θst occurring therein 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 (mμ1s
1s
mμ2s
2s
mμ3s
3s
....) whose partition is a specification of a separation of the function symbolized by (lλ1
1
lλ2
2
lλ3
3
...) is expressible as a linear function of symmetric functions symbolized by separations of (lλ1
1
lλ2
2
lλ3
3
...) and a symmetrical table may be thus formed." It is now to be remarked that the partition (lλ1
1
lλ2
2
lλ3
3
...) can be derived from (mμ1s
1s
mμ2s
2s
mμ3s
3s
....) by substituting for the numbers m1s, m2s, m3s,... 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) = μ1s! μ2s! μ3s! ... Σ (J1)j1(J2)j2 (J3)j3...j1!j2!j3!...
where (J1)j1(J2)j2 (J3)j3... is a separation of (lλ1
1
lλ2
2
lλ3
3
...) of specification (mμ1s
1s
mμ2s
2s
mμ3s
3s
....), placing s under the summation sign to denote the specification involved,
μ1s! μ2s! μ3s! ... ${\displaystyle \sum _{\textit {s}}}$ (J1)j1(J2)j2 (J3)j3...j1!j2!j3!... = ${\displaystyle \sum _{\textit {t=1}}^{\textit {t=k}}}$θst (mμ1t
1t
mμ2t
2t
mμ3t
3t
...)
μ1t! μ2t! μ3t! ... ${\displaystyle \sum _{\textit {t}}}$ (J1)j1(J2)j2 (J3)j3...j1!j2!j3!... = ${\displaystyle \sum _{\textit {s=1}}^{\textit {s=k}}}$ θts (mμ1s
1s
mμ2s
2s
mμ3s
3s
...)
where θst = θts.

Theorem of Symmetry.—If we form the separation function
Σ (J1)j1(J2)j2 (J3)j3...j1!j2!j3!...
appertaining to the function (lλ1
1
lλ2
2
lλ3
3
...), each separation having a specification (mμ1s
1s
mμ2s
2s
mμ3s
3s
....), multiply by μ1s! μ2s! μ3s! ... and take therein the coefficient of the function (mμ1t
1t
mμ2t
2t
mμ3t
3t
....), we obtain the same result as if we formed the separation function in regard to the specification (mμ1t
1t
mμ2t
2t
mμ3t
3t
....), multiplied by μ1t! μ2t! μ3t! ... and took therein the coefficient of the function (mμ1s
1s
mμ2s
2s
mμ3s
3s
....).

Ex.gr., take (lλ1
1
lλ2
2
...)=(214);(mμ1s
1s
mμ2s
2s
...) = (321);(mμ1t
1t
mμ2t
2t
...)=(313); we find

 (21)(12)(1)+(13)(2)(1) =...+13(313)+..., (21)(1)3 = ...+13(321)+...

The Differential Operators.—Starting with the relation
(1 + α1x)(1 + α2x)...(1 + αnx) = 1+a1x+a2x2+...+anxn
multiply each side by 1+μx, thus introducing a new quantity μ; we obtain
(1 + α1x)(1 + α2x)...(1 + αnx)(1+μx) = 1+(a1+μ)x+(a2+μa1)x2+...
so that 𝑓(a1, a2, a3,…an) = 𝑓, a rational integral function of the elementary functions, is converted into
𝑓 (a1+μ, a2+μa1,…an+μan−1) = 𝑓 + d1 + μ22!d2
1
𝑓 + μ33!d3
1
𝑓 + …
where
d1=a1 + a1a2 + a2a3 + … + an−1an
and ds
1
denotes, not s successive operations of d1, but the operator of order s obtained by raising d1 to the sth power symbolically as in Taylor's theorem in the Differential Calculus.

Write also 1s!ds
1
= D, so that
𝑓 (a1+μ, a2+μa1,…an+μan−1) = 𝑓 + μD1𝑓 + μ2D2𝑓 + μ3D3𝑓 + …

The introduction of the quantity μ converts the symmetric function (λ1λ2λ3…) into
(λ1λ2λ3+…)+ μλ1(λ2λ3…)+ μλ2(λ1λ3…) + μλ3(λ1λ2…)+…
Hence, if 𝑓 (a1, a2, a3,…an) = (λ1λ2λ3…),
(λ1λ2λ3+…)+ μλ1(λ2λ3…)+ μλ2(λ1λ3…) + μλ3(λ1λ2…)+…
= (1 + μD1 + μ2D2 + μ3D3 + …)(λ1λ2λ3…).

Comparing coefficients of like powers of μ we obtain
Dλ1(λ1λ2λ3…) = (λ2λ3…),
while Ds(λ1λ2λ3…) = 0 unless the partition (λ1λ2λ3…) contains a part s. Further, if Dλ1 Dλ2 denote successive operations of Dλ1 and Dλ2,
Dλ1Dλ2(λ1λ2λ2…) = (λ1…),
and the operations are evidently commutative.

Also Dπ1
p1
Dπ2
p2
Dπ3
p3
… (pπ1
1
pπ2
2
pπ3
3
… ) = 1, and the law of operation of the operators D upon a monomial symmetric function is clear.

We have obtained the equivalent operations
1 + μD1 + μ2D2 + μ3D3 + … = expμd1
where exp denotes (by the rule over exp) that the multiplication of operators is symbolic as in Taylor's theorem. ds
1
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 = ∂as + a1as+1+a2as+2+….

It has been shown (vide "Memoir on Symmetric Functions of the Roots of Systems of Equations," Phil. Trans. 1890, p. 490) that
exp(m1d1 + m2d2 + m3d3 + …) = exp' (M1d1 + M2d2 + M3d3 + …),
where now the multiplications on the dexter denote successive operations, provided that
exp(M1ξ + M2ξ2 + M3ξ3 + …) = 1 + m1ξ + m2ξ2 + m3ξ3 + ….
ξ being an undetermined algebraic quantity.

Hence we derive the particular cases
expd1 = exp(d112d2 + 13d3 − …);
expμd1 = exp(μd112μ2d2 + 13μ3d3 − …),
and we can express Ds in terms of d1, d2, d3, …, products denoting successive operations, by the same law which expresses the elementary function as in terms of the sums of powers s1, s2, s3, … Further, we can express ds in terms of D1, D2, D3, … by the same law which expresses the power function s, in terms of the elementary functions a1, a2, a3, …

Operation of Ds a Product of Symmetric Functions.—Suppose 𝑓 to be a product of symmetric functions 𝑓1𝑓2…𝑓m. If in the identity 𝑓 = 𝑓1𝑓2…𝑓m we introduce a new root μ we change as into as + μas−1, and we obtain
(1 + μD1 + μ2D2 + … + μsDs + …) 𝑓
=(1 + μD1 + μ2D2 + … + μsDs + …) 𝑓1
×(1 + μD1 + μ2D2 + … + μsDs + …) 𝑓2
×
×(1 + μD1 + μ2D2 + … + μsDs + …) 𝑓m
and now expanding and equating coefficients of like powers of μ
D1𝑓 = Σ(D1𝑓1) 𝑓2𝑓3…𝑓m,
D2𝑓 = Σ(D2𝑓1) 𝑓2𝑓3…𝑓m + Σ(D1𝑓1)(D1𝑓2) 𝑓3…𝑓m,
D3𝑓 = Σ(D3𝑓1) 𝑓2𝑓3…𝑓m + Σ(D2𝑓1)(D1𝑓2) 𝑓3…𝑓m + Σ(D3𝑓1) 𝑓2𝑓3…𝑓m,
the summation in a term covering every distribution of the operators of the type presenting itself in the term.

Writing these results
D1𝑓 = D(1)𝑓,
D2𝑓 = D(2)𝑓 + D(12)𝑓,
D3𝑓 = D(3)𝑓 + D(21)𝑓 + D(13)𝑓,
we may write in general
Ds c = ΣD(p1p2p3…) 𝑓,
the summation being for every partition (p1p2p3…) of s, and D(p1p2p3…) 𝑓 being = Σ(Dp1 𝑓1 )(Dp2 𝑓2 )(D p3 𝑓3 ) 𝑓4…𝑓m.

Ex. gr. To operate with D2 upon (213)(214)(15), we have
D(2) 𝑓 = (13)(214)(15) + (213)(14)(15),
D(12) 𝑓 = (122)(213)(15) + (213)(213)(14) + (212)(214)(14),
and hence
D(2) 𝑓 = (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 (213214)(15).

Write
(213)(214)(15) = … + A(524)(13) + …;
then
D5D4
2
D3
1
(213)(214)(15) = A;
every other term disappearing by the fundamental property of Ds. Since
D5(213)(214)(15) = (13)(14)(14),
we have:—
D4
2
D3
1
(14)(14)(13) = A
D3
2
D3
1
{(13)(13)(13) + 2(14)(13)(12)} = A
D2
2
D3
1
{5(13)(12)(13) + 2(14)(12)(1) + 2(13)(13)(1)} = A
D2D3
1
{12(12)(12)(1) + 7(13)(1)(1) + 2(14)(1) + 6(13)(12)} =A
D3
1
12(1)3 = A,
where ultimately disappearing terms have been struck out. Finally A = 6 · 12 = 72.

The operator d1 = a0∂a1 + a1∂a2 + a2∂a3 + … 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
λ0a0a1 + λ1a1a2 + λ2a2a3 + …
is transformed into the operator d1 by the substitution
(a0, a1, a2, … as, … ) = (a0, λ0a1, λ0λ1a2, … , λ0λ1λs−1as, … ),
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
a0xn − (n
1
) a1xn − 1 + (n
2
) a2xn − 2 − … = 0;
and such functions satisfy the differential equation
a0a1 + 2a1∂a2 + 3a2a3 + … + nan−1an = 0.
For such functions remain unaltered when each root receives the same infinitesimal increment h; but writing xh for x causes a0, a1, a2 a3, … to become respectively a0, a1 + ha0, a2 + 2ha1, a3 + 3ha2, … and 𝑓 (a0, a1, a2 a3, …) becomes
𝑓 + h(a0a1 + 2a1∂a2 + 3a2a3 + …) 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 symmetric functions of the roots of a0xna1xn − 1 + a2xn − 2 − … = 0, are symmetric functions of differences of the roots of
a0xn − 1!(n
1
) a1xn−1 + 2!(n
2
) a2xn−2 − … = 0;
and on the other hand that symmetric functions of the differences of the roots of
a0xn − (n
1
) a1xn−1 + (n
2
) a
2
xn−2 − … = 0;
are non-unitary symmetric functions of the roots of
a0xna11!xn−1 + a22!xn−2 − … = 0.

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
(μ, ν; m, n)
=1m [ μam0
0
αn + (μ + ν)m!(m − 1)! 1!am−1
0
a1αn+1
+ (μ + 2ν) { m!(m − 1)! 1!am−1
0
a2 + m!(m − 2)! 2!am−2
0
a2
1
} αn+2
+ (μ + 3ν) { m!(m − 1)! 1!am−1
0
a3 + m!(m − 2)! 1! 1!am−2
0
a1a2
+