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 vanish. From the above Dpq 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 dpq ＝ $\sum$ars$d⁄a_{p+r,q+s}$ and note the general result $\overline{exp}$ (m10d10 + m01d01 + … + mpqdpq + …) = exp (M10d10 + M01d01 + … + Mpqdpq + …); where the multiplications on the left- and right-hand sides of the equation are symbolic and unsymbolic respectively, provided that mpq, Mpq are quantities which satisfy the relation exp (M10 + M01 + … + Mpq pp + …) = 1 + m10 + m01 + … + mpq pq + …; where, are undetermined algebraic quantities. In the present particular case putting m10 =, m01 = and mpq = 0 otherwise M10 + M01 + … + Mpq pq + … = log (1 + + ) or Mpq = ( − )p+q−1$(p + q − 1)!⁄p! q!$pq; and the result is thus $\overline{exp}$ (d10 + d01) = exp { d10 + d01 − $1⁄2$ (2d20 + 2d11 + 2d02) + … } = 1 + D10 + D01 + … + pqDpq + …; and thence d10 + d01 − $1⁄2$ (2d20 + 2d11 + 2d02) + … =log (1 + D10 + D01 + … + pqDpq + …). From these formulae we derive two important relations, viz. &emsp;( − )p+q−1$(p + q − 1)!⁄p!q!$dpq = $$\sum_\pi$$ ( − )−1 $( − 1)!⁄_{1}! _{2}! …$DD…, &emsp;( − )p+q−1Dpq = $$\sum_\pi$$$(p_{1} + q_{1} − 1)!⁄p_{1}!q_{1}!$1$(p_{2} + q_{2} − 1)!⁄p_{2}!q_{2}!$2 … … $( − )^{{{Greek|Σπ}} − 1)!}⁄_{1}! _{2}! …$dd…, the last written relation having, in regard to each term on the right-hand side, to do with successive linear operations. Recalling the formulae above which connect spq and apq, we see that dpq and Dpq are in co-relation with these quantities respectively, and may be said to be operations which correspond to the partitions (pq), ($\overline{10}$p $\overline{01}$q) 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}!$$1⁄_{2}!$…dd… (multiplication symbolic) corresponds to the partition ($undefined⁄p_{1}q_{1}$1 $undefined⁄p_{2}q_{2}$2…). 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+q−1$(p + q − 1)!⁄p!q!$dpqspq = Dpq($\overline{pq}$) = 1; dpq causes every other single part function to vanish, and must cause any monomial function to vanish which does not comprise one of the partitions of the biweight $\overline{pq}$ amongst its parts.

Since dpq = ( − )p+q−1$(p + q − 1)!⁄p!q!$$d⁄ds_{pq}$ 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 $\overline{pq}$ is absent.

One more relation is easily obtained, viz. $d⁄da_{pq}$ = dpq − h10dp+1,q − h01dp,q+1 + … + ( − )r+shrsdp+r,q+s + ….

—Albert Girard, Invention nouvelle en l’algèbre (Amsterdam, 1629); Thomas Waring, Meditationes Algebraicae (London, 1782); Lagrange, ''Mém. de l’acad. de Berlin (1768); Meyer-Hirsch, Sammlung von Aufgaben aus der Theorie der algebraischen Gleichungen (Berlin, 1809); Serret, Cours d'algèbre supérieure,'' t. iii. (Paris, 1885); Unferdinger, ''Sitzungsber. d. Acad. d. Wissensch. i. Wien,'' Bd. lx. (Vienna, 1869); L. Schläfli, “Ueber die Resultante eines Systemes mehrerer algebraischen Gleichungen,” Vienna Transactions, t. iv. 1852; MacMahon, “Memoirs on a New Theory of Symmetric Functions,” American Journal of Mathematics, Baltimore, Md. 1888–1890; “Memoir on Symmetric Functions of Roots of Systems of Equations,” ''Phil. Trans.'' 1890.

A binary form of order n is a homogeneous polynomial of the nth degree in two variables. It may be written in the form ax + bx'x2 + c'x + …; or in the form ax + bx'x2 + c'x + …; which Cayley denotes by (a, b, c, …) (x1, x2)n , … being a notation for the successive binomial coefficients n, n (n − 1), …. Other forms are ax + nb'x2 + n(n − 1)cx'x + …, the binomial coefficients being replaced by s!, and ax + $1⁄2$b'x2 + $1⁄1!$cx'x + …, the special convenience of which will appear later. For present purposes the form will be written $1⁄2!$x + $undefined⁄a_{0}$x'x2 + $undefined⁄a_{1}$x'x + … + $undefined⁄a_{1}$x, 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 $undefined⁄a_{n}$, $undefined⁄a_{0}$, $undefined⁄a_{1}$, … $undefined⁄a_{2}$, n + 1 in number are arbitrary. If the form, sometimes termed a quantic, be equated to zero the n + 1 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(x1, x2) be subjected to the linear transformation x1 = 111 + 122, x2 = 211 + 222, 1, 2 being new variables replacing x1, x2 and the coefficients 11, 12, 21, 22, termed the coefficients of substitution (or of transformation), being constants, we arrive at a transformed quantic 𝑓 (1, 2) = $undefined⁄a_{n}$ + $\overline{a}$2 + $\overline{a}$2 + … + $\overline{a}$ in the new variables which is of the same order as the original quantic; the new coefficients $\overline{a}$, $\overline{a}$, $\overline{a}$ $\overline{a}$ are linear functions 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 r1 = 22x1 − 12x2, r1 = − 21x1 − 11x2, where r =  = 1122 − 1221; r is termed the determinant of substitution or modulus of transformation; we assure x1, x2 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($\overline{a}$, $\overline{a}$, $\overline{a}$, … $\overline{a}$) = r&#8202;undefined F($\overline{a}$, $undefined⁄a_{0}$, $undefined⁄a_{1}$, … $undefined⁄a_{2}$), the function F($undefined⁄a_{n}$, $undefined⁄a_{0}$, $undefined⁄a_{1}$, … $undefined⁄a_{2}$) is then said to be an invariant of the quantic quâ linear transformation. If, however, F involve as well the variables, viz. F($undefined⁄a_{n}$, $\overline{a}$, $\overline{a}$, … ; 1, 2) = r&#8202;undefined F($\overline{a}$, $undefined⁄a_{0}$, $undefined⁄a_{1}$, … ; x1, x2), the function F($undefined⁄a_{2}$, $undefined⁄a_{0}$, $undefined⁄a_{1}$, … ; x1, x2) 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 ($undefined⁄a_{2}$, $undefined⁄a_{0}$, $undefined⁄a_{1}$, … ; x1, x2), ($undefined⁄a_{2}$, $undefined⁄b_{0}$, $undefined⁄b_{1}$, … ; 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 ($undefined⁄b_{2}$, $\overline{a}$, $\overline{a}$, … ; 1, 2), ($\overline{a}$, $\overline{b}$, $\overline{b}$, … ; 1, 2), … If then we find F($\overline{b}$, $\overline{a}$, $\overline{a}$, … $\overline{a}$, $\overline{b}$, $\overline{b}$, …, …; 1, 2), = r&#8202;undefined F($\overline{b}$, $undefined⁄a_{0}$, $undefined⁄a_{1}$, … $undefined⁄a_{2}$, $undefined⁄b_{0}$, $undefined⁄b_{1}$, …, …; x1, x2),