Page:The New International Encyclopædia 1st ed. v. 06.djvu/207

* DETERMINANTS. 16ii DETERMINATE PROBLEM. ^^iUh^c, observed that A,, 1?,,. . . invulve their own signs, and that in the above equation 15, and D, are negative. Thus a determinant of the iitli order may be expanded in terms of determinants of the (n — l)th onler. and so on. Determi- nants also admit, in their abridged form of no- tation, of the fundamental operations of addi- tion, subtraction, and multiplication (including involution). These functions are of great importance in the solution of simultaneous equations. The roots in the case of two linear equations, a^x + b,y = c, and a..j! + 6j/ = C:a »re expressed thus: a; = (C6:) (aiC;) ^ ^,, ,. J — i— ;' y:= ', — j—, In the case of tliree equa- tions involving three unknowns, of the type + ^ly + Ca~- = ''a- the roots are _ A/i^.cA / a,d^, ( iiMs ) In the case of n linear equations involving n unknowns, , A-n) (ai6:C3. . . fcn)' For n 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 a^x + b,y + c,c; = 0, a,x+ bj/ + c^ = 0, and a^x + b^y + c^z = 0, if (ajft-c,) = 0, the equations can be satisfied by a set of values of x, ii. ::. This determinant of the coe.lieients is called the discriminant or the elimi- nant 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 be expressed in determinant form. Thus the eliminant of two equations, one of the )»th and the other of the jith degree, may be found by a method known as Sylvester's diahjUc process; e.g. to form the eliminant of px^ + 70; -f r = and ax' + bx- + cor + (Z = 0. multiply the members of the first equation by x and by j-, and the second by j, and form a system of five equations considering x', x', ^, and X as the unknowns. The eliminant is p q r a b c d a b c i1 =0. /) r; )• p q r w are functions of x, y, ::, the deter- If u, minant /dn I dx ' dv dw ~d: ic d 7)= d d(ii. -^ = J{uv,c) Mx, y, IS called the Jacobian of the system h. r. ir, with respect to x. y, z. In the particular case where «, 1;, ir are the partial difTcrential co- efficients (see CALCULU.S) of the same function of the variables x, y, z, it is called the Hessian of the primitive function. These and other ex- pressions play an important part in expressing the properties of certain cun-es (q.v.) and sur- faces : e.g. it is sliown in modem geometry that if the first polar of any point A, with respect to a curve « =: 0, homogencois in x. y, z. has a double point U, the polar conic of B has a double point. The locus of the double point B is expressed by the Hessian I" '' 9 h b f = 19 f c which is satisfied by x, y, :, and in which iJ, 6, c . . . are diirerential coefiicients of the second order. This function is also an example of a cova riant. See Forms. The idea of determinants may be said to take its origin with Leibnitz (11)93), following whom Cramer (1750) added somewhat to the theory, treating the sulijects wholly in relation to sets of equations. The recurrent law was first an- nounced by Bezout (1704). But it was 'andcr- nionde (1771) who first recognized determinants as independent functions, 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 determi- nant in terms of its complementary minors. Ira- mediately following, Lagrange (1773) treated determinants of the second and third orders, being the first to apply these functicms 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 (1812) the theory of determinants 1)egins in its generality. The next great contril)utor, and the greatest save Cauchy, was .Jacobi (from 1827). With him the word determinant received its final acceptance. He early used the functional determinant, which Sylvester has called the .lacobian, and in his famous memoirs, in Crelle for 1841, he specially treats this subject, as well as that class of alternating functions known as alternants. But about the time of .Lacobi's clos- ing memoirs, Sylvester (1839) 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- ric determinants have been studied by Lebesgue, Hesse, and Sylvester: pcr-symmetric determi- nants by Hankel : circulants by Catalan, Spot- tiswoode, CJlaisher, and Scott; skew determi- nants and Pfaft'ians, in connection with the theory of orthogonal transformation, by Cayley; continuants by Sylvester; Wronskians by Chris- totfel and Frobenius ; Jacobians and Hessians by Sylvester: and symmetric gaiiche determi- nants by Trudi. The theoiy as a whole has been most systematically treated by F. Brioschi ( 1824- 1897), well known as the editor of the Annali di Matematica, whose masterly treatise on deter- minants is a standard. (Frencli and German translations. ISoO. See also his Opcrc mate- matiche. Milan. 1901 — .) Consult: Jluir. Theory of Determinants in the Historical Order of Its De- velopment, part i. (London, 1890) ; Baltzer. 7Viro- rie und Anircndungen der Determinanten (Leip- zig, 1881); Doster, EUnients de la theorie des determinants (Paris, 1877) ; Scott, Theory of Determi)iants (Cambridsrc, ISSO) ; Salmon. Les- sons Introdurtory to the Modern Higher Alyebra (Dublin. 1S7C) ; Mcrriman and Woodward, lliyhcr Ualheniiities (New York, 1808). DETERMINATE PROBLEM. In algebra or geometry, a problem of a limited number of solutions : e.g. given the base, |)criineter, and area of a triangle to construct it. In this prob- lem there are, in general, four solutions. But if