MATHEMATICS, RECENT TERMINOLOGY IN. The terms intended to be explained in the present article relate to subjects distinct indeed, but intimately connected together, as well logically as historically. Determinants were devised as a means to the solution of a system of simple equations, but the principle of their construction is contained in the rule of signs which belongs to the theory of arrangements (or permutations): this theory has been studied, as well for its own sake, as in reference to the theory of equations, and in it originated the notion of a group, the most outlying term of those which are here explained. Moreover, in a system of simple equations, if the coefficients arranged in the natural square order are considered apart by themselves, this leads to the theory of matrices, a theory which indeed might have preceded that of determinants ; the matrix is, so to speak, the matter of a determinant; the rule of signs giving the form. But when the rule of signs is applied to other matter, this leads to the functions called permutants ; these include eommutants and intermutants, and also Pfaffians, which however were not origi nally so arrived at. The theory of elimination (according to one of the two ways in which it may be treated) is essentially dependent upon systems of linear equations, and is thus also connected with deter minants. And all, or nearly all, the before-mentioned theories have an application to the theory of rational and integral homogeneous func tions, or, as they have been termed, forms or panties ; they are thus connected with the "Calculus of Forms," and with " Quanties ;" the last-mentioned expression (used as a singular), has been defined to denote the entire subject of rational and integral functions, and of the equations and loci to which these give rise. The theory of rational and integral functions was first studied in a general manner in the question of linear transformations, and it was this question which led to the discovery of the functions, called originally hyperdderminanta, but afterwards invariants, and of the more general functions called covariant* : the theory of comuiants is indeed the part which has been chiefly attended to of the Calculus of Forms, or of Quantiai.
Ruts or SIONS.—Any arrangement of a series of terms may bo derived (and that in a variety of ways) from any other arrangement by succesnivo interchanges of two terms ; but in whatever way this is done, the number of interchanges will be constantly even, or else constantly old; and the two arrangements are said to be of the same sign or of contrary signs accordingly. In particular, if any arrange ment is selected as the primitive arrangement, and taken to be positive, then any other arrangement will be positive or negative according as it is derivable from the primitive arrangement by an even or an odd number of interchanges. The definition leads to the following theorem : any arrangement is positive or negative, according as the total number of times in which the several elements respectively precede (mediately or immediately) elements posterior to them in the primitive arrangement, is even or odd : it may be added, that tho positive and negative arrangements are equal in number. Thus in the case of three terms,
the primitive arrangement being 123; the positive arrangements are 123, 231, 312, the negative arrangements, 132, 218, 321: in the case of four terms, the primitive arrangement being 1234, tho arrange ments 1234, 2341, 3412, 9123 are respectively positive, negative, peal tive, negative ; there are in all twelve positive and twelve negative arrangements.
Cnour.—The term was originally used as applied to substitutions only, but the more general use of the term is as follows : let 0 bo a aymbol operating on any number of terms a., y a .. and producing as the result of the operation the same number of new terms x, v, 2, .. (where x, T, s ... may be each of them functions of all or any of the ect, x, y, z . ; if a, r, z .. are merely the terms, x, y, z .. in a different order, then 0 is a substitution, which explains in what sense that term has just been used). Imagine a set of operative symbols, 1, 0, X • • (1, as an operative symbol denotes, of course, n symbol which leaves, the operand unaltered) such that the result of the operation of any twonymbeln 0, is (the same or and if different, then in either order) is identical with that of the operation of some symbol x of the set ; AS thus, 00(x, y, ..)= (x, v, 2, ..)= I", ..)=x(r, y, a..), &V, ; then the symbols 1, Op X . • . form a group. It is to be remarked that 1 belongs to every group, and moreover, that if 0 be any symbol of the group, then 8', belong to the group : the most annplo form of a group (and when the number of terms is prime, the only form) is 1, 0, . . . [0 3loro generally, if there are n terms in the group, then every symbol 0 of the group is an operation 'periodic of the order a (if not of an order a nulimultiple of n) and thus satisfies the eymbolic equation Os =1. The symbols of the group are, so to speak, the symbolic it-th roots of unity, and as in the above mentioned example, they may, whether a is pnme or composite, form a Teem precisely anelogoua to that of the ordinary n-th roots of unity ; but when a is composite, then upon two grounds this is not of necessity the case. 1'. The symbols of a group need not be converti ble (thus a=s., there Is a group, Calm sa= air and .*. Ira aa, this is in fact, the group of the substitution,' of three timings). 2". There may be distinct n-th roots, thus n = 9, there is a group, 1,a, iit,a0 [al= idri=i,a.8=ftal in which a, LI are distinct quare root. of (the symbolical) unity, and which is thus wholly lifferent from the group, 1, a, a', a' = 1).
The combination of a series of terms in the way of addition or sub raction, according to the rule of signs, gives rise to the chum of functions gilled permidante, which include ns n particular but the earliest din iovered and moat important case, the determinant