Percy Alexander MacMahon

“ Symmetric Functions of Several Systems of 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
we start with the fundamental relation
As shown by L. Schläfli [2] 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. ”

“ 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. ”

“ The quantities are called the elements of the determinant; the term is called a member of the determinant, and there are evidently members corresponding to the permutations of the numbers The determinant is usually written
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 rows and columns. ”

“ 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. ”

“ Operation of upon a Product of Symmetric Functions.—Suppose to be a product of symmetric functions If in the identity we introduce a new root we change into and we obtain
and now expanding and equating coefficients of like powers of
the summation in a term covering every distribution of the operators of the type presenting itself in the term.
Writing these results
we may write in general
the summation being for every partition of and being
Ex. gr. To operate with upon we have
and hence
Application to Symmetric Function Multiplication.—An example will explain this. ”

“ Now , wherein is not to be changed, but the second suffixes in the product assume all permutations, the number of transpositions necessary determining the sign to be affixed to the member.
Hence , where the cofactor of is clearly the determinant obtained by erasing the first row and the first column.
Hence
Similarly , the cofactor of , is shown to be the product of and the determinant obtained by erasing from the i th row and k th column. No member of a determinant can involve more than one element from the first row. ”

“ Putting and we find a particular law of reciprocity given by Cayley and Betti,
and another by putting , for then becomes , and we have
Theorem of Expressibility.—“If a symmetric function be symboilized by and be any partitions of respectively, the function is expressible by means of functions symbolized by separation of
For, writing as before,
is a linear function of separations of of specification and if is a linear function of separations of of specification Suppose the separations of to involve different specifications and form the identities
where is one of the specifications. ”

“ Expanding the right-hand side by the exponential theorem, and then expressing the symmetric functions of which arise, in terms of we obtain by comparison with the middle series the symbolical representation of all symmetric functions in brackets appertaining to the quantities To obtain particular theorems the quantities are auxiliaries which are at our entire disposal. ”

“ Thus to obtain Stroh’s theory of seminvariants put
we then obtain the expression of non-unitary symmetric functions of the quantities as functions of differences of the symbols
Ex. gr. with must be a term in
and since we must have
as is well known.
Again, if be the roots of and leading to
and
and we see further that vanishes identically unless . If be infinite and
we have the symbolic identity
and
Instead of the above symbols we may use equivalent differential operators. Thus let
and let be equivalent quantities. ”

“ At various times he was president of the Cambridge Philosophical Society, of the London Mathematical Society and of the Royal Astronomical Society. He was elected a fellow of the Royal Society in 1852, and received from that body a Royal medal in 1859 and the Copley medal in 1882. He also received the De Morgan medal from the London Mathematical Society, and the Huygens medal from Leiden. ”

“ American 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
axn
1 + bxn−1
1x2 + cn−2
1x2
2 + ...;
or in the form
axn
1 + (n
1) bxn−1
1x2 + (n
2) cn−2
1x2
2 + ...;
which Cayley denotes by
(a, b, c, ...) (x1, x2) n
(n
1) , (n
2) ... being a notation for the successive binomial coefficients n, 1/2n (n − 1) , .... ”

“ CAYLEY, ARTHUR (1821–1895) , English mathematician, was born at Richmond, in Surrey, on the 16th of August 1821, the second son of Henry Cayley, a Russian merchant, and Maria Antonia Doughty. His father, Henry Cayley, retired from business in 1829 and settled in Blackheath, where Arthur was sent to a private school kept by the Rev. ”

“ Any function of differences of being formed, the expansion being carried out, an operand or or being taken and being subsequently put equal to , a non-unitary symmetric function will be produced.
Ex. gr. ”

“ Since
dpq = ( − ) p+q−1 (p + q − 1) !/p!q!d/dspq
the solutions of the partial differential equation dpq = 0 are the single bipart forms, omitting spq, and we have seen that the solutions of Dpq = 0 are those monomial functions in which the part pq is absent.
One more relation is easily obtained, viz.
d/dapq = dpq − h10dp+1,q − h01dp,q+1 + ... + ( − ) r+shrsdp+r,q+s + ... .
References for Symmetric Functions.—Albert Girard, Invention nouvelle en l’algèbre (Amsterdam, 1629) ; Thomas Waring, Meditationes Algebraicae (London, 1782) ; Lagrange, Mém. de l’acad. de Berlin (1768) ”