RATIONAL AND INTEOILAL FUNCTIONS (Notation and Nomenclature of).—A rational and integral homogeneous function, such as the function 26.ry is denoted by (*) (x, where the coefficients are only indicated by the asterisk, but are not expressed. A non-homogeneous rational and integral function is con sidered as derived from a homogeneous function by putting one of the variables thereof equal to unity, and is represented accordingly : thus axe + 2bx + c is denoted by (*) (x, But it Is often proper to express the coefficients, and in regard to this the following distinction is made, namely— (a, 6,c)(x, denotes a + 2bxy+ ; and in like manner (a, b, r, d) (x, y)' denotes ea-3+3 b + 3 cxyl+ d ys, &c., the numerical coeffictenta being those of the successive powers of a binomial. But when such numerical coefficients are not to be inserted, this is denoted by an arrowhead, or other distinctive mark ; thus (a, b, denotes a xi +6 x y + c y'. A rational and integral function of any order is termed a quantic, and a function of the orders two, three, four, five, &c., is termed a quadric, cubic, quartie, quintie, respectively. The number of variables (the function being homogeneous) is denoted by the words binary, ternary, &c. As a correlative term to coefficients, the variables have been termed laciest:. A function which is linear in respect to several distinct seta of variables separately is said to be tantipartitc : or, when there are two seta only, linco-lincar. Thus a determinant is a tantipartite function of the lines or of the columns; the function axx'+bxy'+cey+dyy' is a lineo-linear function of (x,y) and(x', y') ; a notation for it is ( a, b)(x, I a' ,b' I such as has been spoken of in regard to matrices. EXANANT.—The development of an expression such as (* )(A x +s Ay is naturally written under the form (*) (x, Y)' + T (*)(x, y)"' i) + ce, :0" #4`, and the coefficients of the successive terms A', &c., are raid to be the emanants of the quantic (*) The coefficients of or 0-th cmanant,is the quantic itself, and the coefficient of le', or ultimate emanant, is the quantic with (x', y') in the place of (x, y) ; but the intermediate emanants are functions of (x, y) and (x', y') homogeneous in respect to the two seta separately. The coefficients may of course be expressed thus, the emanant (first emanant) of (a, 6, c) (x, is (a, b, c) (x, y) (e, 0, which stands for a zal + b y' + y)+ cy Lniesn TRANSFOnMATIoNs.—In this theory the variables of a function
are supposed to be respectively linear functions of a new set of variables, so that the function is transformed into a similar function of these now variables, with of course altered values of the coefficients, and the question was to find the relations which existed between the original and new coefficients and the coefficients of the linear equations. The determinant composed of the coefficients of the linear equations is said to be the modulus of transformation, and when this determinant is unity the transformation is said to be unimodular. It was observed that a certain function of the coefficients, namely, the discriminant, possessed a remarkable property, found afterwards to belong to it as one of a class of functions called originally hyperdeterminants, but now invariants, and it was in this manner that the problem of linear trans formation led to the general theory of covarianta.
IevaniaNr.—An invariant is a function of the coefficients of a rational and integral homogeneous function or quan tic, the characteristic I property whereof is as follows : namely, if a linear transformation is effected on the quantic, then the new value of the invariant is to a factor prPs equal to the original value; the factor in question (or quotient of the two values) being a power of tho modulus of transfer mation, and the two values being thus equal when the transformation is unimodular. The easiest example is afforded by the quadric function (a, b, c) (x, y)' ; effecting upon it a linear transformation, suppose that we have identically c) (a = b' e') then it may be easily verified that a' The invariant a c—b' Is however in this case nothing else than the dia. criminant ; as another example take the quartio (a, 6, c, d, e) (x, y)•, for which a c-4 b cl+ Sc', a c c—a + 2b cd—cd are functions possessed of the like property of remaining to a factor prh unaltered by the transformation, and are consequently invariants ; it may be added that calling them r, J, respectively, the discriminant is here =P-27 a rational and integral function of invariants of a lower degree.