These functions are of great importance in the solution of simultaneous equations. The roots in the ease of two linear equations, a,x+ b,y = and ax = c„,, are expressed thus: x = (e,b,) (a b..,) In the case of three equa dons involving three unknowns, of the type a,x b,y =d,, a„ .r b,y e = (I,. +b,y = the roots are a,b, )' In the ease of n linear equations involving It unknowns, —(/, ... (tlike3 • • • For a homogeneous linear equations, the determi nant notation serves to express the necessary and sufficient condition for the consistency of the sys tem. Thus, in the system b,y = 0.
(kse = 0, and a,..rb,y c,z = 0, if = 0, the equations can be satisfied by a set of values of y, z. This determinant of the coefficients is called the discriminant or the elimi swat of the system of equations. It is the ex pression which results from eliminating the un knowns from the given equations. The discrimi nants of higher equations also may he expressed in determinant form. Thus the eliminant of two equations, one of the filth and the other of the nth degree, may be found by a method known as Sylvester's dialytie process; e.g. to form the eliminant of r = 0 and ax' d = O. multiply the members of the first equation by •r and by X', and the second by a% and form a system of five equations considering x', at', and x as the unknowns. The eliminant is oOptic O a b e d a b e (I 0 =O.
O p q r 0 pqr 0 0 If u, v, w are functions of y, z, the deter minant (du dv due d( u. V. ar) dy' d:11(x, y, z.) = is called the Jaeobian of the system a. r, with respect to x, y, z. In the particular ease where u, r, ic are the partial differential co efficients (see ) of the same function of the variables a-, y, a', it is called the Hessian of the primitive function. These and other ex pressions play an important part in expressing the properties of certain curves (q.v.) and sur faces: e.g. it is shown in modern geometry that if the first polar of any point -A, with respect to A curve a = 0, homogeneous in X, N. Z. a double point IL the polar conic of B has a double point A. The loons of the double point B is expressed by the Hessian a h g h b fi= 0 f e which is sot islied by x, y, and in which a, b, e . . are differential coefficients of the second order. This function is also an example of a covariant. See Foams.
The idea of determinants may be said to take its origin with Leibnitz (1693). following whom Cramer (1750) added somewhat to the theory, treating the subjects wholly in relation to sets of equations. The law was first an nounced by 11,'Izont (1764). But it was Vander monde (1771) who first recognized determinants as independent funetions, giving a connected ex position of the theory, and hence he deserves to be called its formal founder. Laplace (1772)
gave the general method of expanding a did ermi mint in terms of its complementary minors. Im• mediately following, Lagrange (1773) treated determinants of the second and third orders, being the first to apply these functions to ques tions foreign to eliminations, and he discovered many special properties. Gauss (1801) intro duced the name determinants, although not in its present sense; he also arrived at the notion of reciprocal determinants, and came very near the multiplication theorem afterwards given by . Binet (1811-12) and Cauchy. With the latter (ISM the theory of determinants begins in its generality. The next great contributor, and the greatest save Cauchy. was Jacobi (from 1S27). With him the word determinant received its final acceptance. He early used the functional determinant, which Sylvester has called the Jacobian, and in his famous memoirs. in Crelle for 1841, he specially treats this subject, as well as that Blass of alternating functions known as alternants. But about the time of Ja•obt's clos ing memoirs, Sylvester (1830) and Cayley (qq.v.) began their great work, a work which it is im possible to summarize briefly, but which repre sents the development of the subject to the present time. The study of special forms of determinants has been the natural result of the completion of the general theory. Axi-symmet rie determinants have been studied by Lebesgue, Hesse, and Sylvester: per-symmetrie determi nants by Hankel ; circulants by Catalan, Spot tiswoode. Glaisher, and Scott; skew determi nants and Pfaffians, in connection with the theory of orthogonal transformation, by Cayley; continuants by Sylvester; Wronskians by Chris toffel and Probenius; Jaeobians and Hessians by Sylvester: and symmetric gauche determi nants by Truitt The theory as a whole has been most systematically treated by F. Brioschi (1824 1897), well known as the editor of the Annan di Matemat iea, whose masterly treatise on deter minants is a standard. (French and German translations, 1856. See also his Opere mate matiehe, 1901—.) Consult: Aluir, Theory of Determinants in the Historical Order of its De velopment, part i. (London, 1890); Balt wt., Theo vie and .1 nmendunyen der Determinanten (Leip zig, 1881) ; Duster. EWments dr ht thl'o•ie des daerminants (Paris. 1877) ; Scott. Theory of Determinants (Cambridge, 1880) : Salmon. Us xnas int•odurtory to the Modern Higher .11yebr« (Dublin, 1876) ; and Woodward, Higher .1Iathematies (New York, 1808),