MATHEMATICS, RECENT TERMINOLOGY IN. 



MATHEMATICS, RECENT TERMINOLOGY IN. US 



bat aftrrwardu inmnoMif, and of the more general functions called 

 nmnamu : the theory of oovariants is indeed (he part which has been 

 chiefly tW*~< to of the Calculus of Form*, or of Qualities'. 



The following list of term* may be convenient : 



Rule of signs. 



Group. 



Determinant. 



Minor, 



Symmetric, skew, skew symmetric. 

 Commutant. 



IVrmutant. 



Intel-mutant 



Cumulant 



Matrix. 



Resultant 



Discriminant 



Plexus. 



Ratytv' and integral functions (notation and nomenclature of). 



Quontic, quadric, cvibic, &c., binary, ternary, 4c., facient, tanti 



partite, linoolinear. 

 Emanant 

 Linear transformations. 



Modulus of transformation, unimodular. 

 Hyperdeterminant 

 Invariant. 

 Covariant 



Contravariant, peninvariant, scminvariant, quadrinvariant, qua- 



ilricuvariant, &c., catalocticant, canonisant, 

 Canonical form. 

 Bczoutic matrix, &c. 

 Tactinvariant, reciprocant 

 Functional determinant, Jocobian, Hessian. 

 Concomitant, cogrcdient, contragredienc. 

 Combinant 



RVLE OF SIGNS.- Any arrangement of a scries of terms may be 

 derived (and that in a variety of ways) from any other arrangement by 

 successive interchanges of two terms ; but in whatever way this is 

 done, the number of interchanges will be constantly even, or else 

 constantly odd ; 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 o 

 times in which the several element* respectively precede (mediately or 

 immediately) elements posterior to them in the primitive arrangement 

 is even or odd: it may be added, that the 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, 182, 213, 321 : in the case 

 of four terms, the primitive arrangement being 1234, the arrange 

 ments 1284, 2341, 8412, 4123 are respectively positive, negative, posi 

 tire, negative ; there are in all twelve positive and twelve negative 

 arrangements. 



QBOUP. The term was originally used as applied to substitution 

 only, but the more general use of the term is as follows : let bo a 

 symbol operating on any number of terms x, y z . . and producing a. 

 the result of the operation the same number of new terms s, T, z, . 

 (where x, T, s . . . may be each of them functions of all or any of th 

 set, z, ;/, r . . ; if x, T, z . . are merely the terms, x, y, z . . in a diffcren 

 order, then 6 is a substitution, which explains in what sense that term 

 has just been used). Imagine a set of operative symbols, 1, 9, <f>, x 

 (1, as an operative symbol denotes, of course, a symbol which leaves 

 the operand unaltered) such that the result of the operation of any 

 two symbols 0, 9 (the same or different, and if different, then in oithc 

 order) is identical with that of the operation of tome symbol x of thi 

 set; as thus, ?(*., i . .) = 0<x, T, z, . .)=(x', Y 7 , t", . .) = x(*. ?,*) 

 ay, *= x ; then the symbols 1, 9, 0, x form o yrmp. It is to be 

 remarked that 1 belongs to every group, and moreover, that if 9 bo any 

 symbol of the group, then f, *,,.. belong to the group : the mos 

 simple form of a group (and when the number of terms is prime, the 

 only form! is 1, ,*... -' [ !]. More generally, if there are n 

 terms in the group, then every symbol 8 of the group is an operation 

 periodic of the order * (if not of an order a submultiple of n) and 

 thus satisfies the symbolic equation fl " = 1 . The symbols of the group 

 are, so to speak, the symbolic n-th roots of unity, and as in the above- 

 mentioned example, they may, whether n is prime or composite, form 

 a system precisely analogous to that of the ordinary n-th roots o 

 unity ; but when is composite, then upon two grounds this is not o 

 necessity the case. 1*. The symbols of a group need not be convorti 

 ble(thus = 6, there is a group, l,ft F,a,a0,a0> [a'=l,0=l,0a^ 

 and .'. 8* = a, thin is in fact, the group of the substitutions o 

 three thing.). 2'. There may be distinct -th roots, thus n = I 

 is a group, I,a,0,o0 [a=l,/3=*l,a3=/3a], in which a, are distinc 



square roots of (the symbolical) unity, and which is thus wholly 

 Ifferent from the group, 1, o,o*,o [!]. 



The combination of a series of terms in the way of addition or MI!>- 

 traction, according to the rule of signs, gives rise to the clam of f unctions 

 called permutanU, which include as a particular but the earliest dis- 

 covered and most important case, the determinant. 



DmmtkUXAKT. Imagine a square arrangement of terms, for 

 ixample 



a, ft, e 



and taking this as the primitive arrangement, permute in every possible 

 way entire columns (or, what would give the same result, entire lines) 

 and for each such arrangement form the product of the terms in the 

 dexter diagonal (.w to B.K) of the square, giving to such product the 

 sifsn which belongs to the arrangement of the columns (or lines). 

 The algebraical sum of these products is a determinant, and such 

 determinant is, or may be represented as above, by inclosing the terms 

 within two vertical lines. Thus the developed value of the determinant 

 in question is 



a4'e'-a6V+a'4"c+a"&e'-a"4'c-a'6"c 



The rule may be otherwise stated as follows : a determinant is the sum 

 of a series of products each with its proper sign,sucu that in each product 

 the factors are taken out of each line and out of each column, and if the 

 factors are arranged according to the primitive arrangement of the 

 columns in which they occur, then the sign is that corresponding to 

 the resulting arrangement of the lines (or vice rcrsd) ; thus in the 

 product oi" c", the factors a,b",c' occur in the columns 1, 2, 3 (they 

 are therefore arranged according to the primitive arrangement of the 

 columns) and in the lines 1, S, 2. Such arrangement of the lines con- 

 sidered as derived frdm the primitive arrangement 1 2 3 U negative, 

 and the product has therefore the sign. A generalisation of this 

 construction will be mentioned under the term commutant. 



The word rttultant was formerly used as synonymous with determi- 

 nant, but it is now employed and is hero explained in a more extended 

 signification. The new synonym tliminant seems unnecessary. 



A few of the numerous properties of determinants may be stated. 



A determinant is a linear function (without constant term) of the 

 terms in each of its columns, and also of the tenns in each of its lines, 

 or, more brieny expressed, it is a linear function of each column, and 

 also of each line. Moreover, without altering the value of tlu 

 minant, the lines may be made columns, and the columns lines, and 

 all the properties of the function exist equally with respect to the 

 lines and with respect to the columns. The absolute value of the 

 determinant is not altered, but the sign is reversed, by an interchange 

 of two columns, hence also if two columns become identical, the 

 determinant vanishes. Moreover when the columns are permuted in 

 any manner whatever, the absolute value is not altered, but the sign 

 will be that corresponding to the arrangement of the columns. A 

 determinant may be developed as a linear function of the terms in any 

 line, thus 



V, 



I <', ' I + c 



tho signs being alternately positive and negative or else all positive, 

 according as the number of columns is even or odd. 



The square arrangement of terms out of which a determinant is 

 formed, and generally any square or rectangular arrangement of terms, 

 is called a matrix. Consider a determinant. 



a, b, c, d 

 a', b', c', d' 

 o||, W, c", t d" Hi 



and partitioning tho linos in any manner, form with them tho matrices 

 \a,b,e,d\ \ a", b", c", d" 



/,'/,' S A! \ > n"' I,'" f"' if" 



\a,o,c,a | | a ,o , c ,a 



anil out of these matrices, with complementary columns thereof, n sum 

 of products 



- . I a, h \ \c", d" 

 6 i | a', V \ \ c"', d'" ' 



(tho sign + being that corresponding to the product a b'c" d"' of tho 

 terms in the dexter diagonals of the factor determinants, considered as 

 a term of tho original di-U-rminant), tho sum of all the products so 

 obtained is the origin >l determinant 



It has been mentioned that tho determinant is a linear function of 

 each column ; hence if the terms of any column are pa, pa' . . tho 

 determinant is equal to f times a determinant in which the corres- 

 ponding column if a, a' ... and similarly if tin i-olnmn ia + b,a' + b', .. 

 then tho determinant if tho sum of two other determinant* in which 

 the corresponding columns are a, a',... and 4, 4,' ... respectively. 



