SUBSTANTIVE LAW. 652 STJBTKACTION. conditions established by the substantive law, and the rectification of such abnormal social conditions as may arise. For convenience, the substantive law is subdivided into various branches according to the subject matter to which it relates. For example, we speak of the law of real property, of personal property, the law of domestic relations or persons, etc. It is to be found both in legislative enactments and in the rules and precedents of the common law. See Law and the authorities there referred to. SUBSTITUTION (Lat. siihstitiitio, a putting in the place of another, from substitiiere, to put in the place of another, from sub, under + statuerc, to place, from stare, to stand). A mathematical operation hj which one expression is replaced by another. The term has, however, come to have a teclinical meaning in modern mathematics, and this has led to an important branch known as the theory of substitutions. If n elements, «,, a^, a,, . . . a^ are given, and a-,, O;, «:,...(/„ and a,, a-i, a,, . . . o^ are two ar- rangements of these elements, the operation of passing from the first of these arrangements to the second is called a substittition of the n ele- ments. It follows that there are no substitutions of »■ elements, including the identical substitu- tion, which leaves the order of the letters un- changed. A substitution which in place of the arrangement o,, «,, o,, ... n^ gives a, a'., o'„ .. .a, is represented by the symbol /«1 "2 O3 • • • • «n "X a a'j o's . . . . a,J If, however, a-^ is replaced by a,, Oj by a,, . . . a„_i by o„, and a„ by a,, the substitution is said to be cyclic, and is inore conveniently repre- sented by (Oi, Oj, (h- ■ ■ •"„), or even by (1, 2, 3, ... n), than by the more elaborate symbol ("1 "2 . ■ ■ • o„-i a„ Similarly a substitution like [cedafb ) may be written {acd) (bef), meaning that while a changes to c. c to (I. and d to a, 6 at the same time changes to c, e to f, and f to 6. This sjtu- bolism is further extended thus: Consider {ab) (oc) ; this means that a changes to b, b to n, and a to c, and c back to 0, a result which evi- dently may also be indicated by {abc) , so that {ab)'(ac)'— (abc). But the same reasoning shows that {ac) {ab) = (nc6). Hence if s, = (ab) and S:=(f/e), SiSj ± s,Si. For conveni- ence, SjSj is called the product of s^ and s, in the order given, from which it appears that the commutative law of multiplication does not hold true in the theory of substitution. If in the product S1S0S3 • • ■ s„ we have Si = Sj := ... s, , the product is called the power of each substitution. If a substitution leaves all the elements un- changed in order it is called an identical substi- tution and is represented by 1. If the product of two substitutions, like A'l "2 •••"9') and ('""'=■ •■"-') is I, each is called the inverse of the other, and if the first is represented by s, the second is represented by s~^, ss~^ equaling 1. A collection of substitutions is said to form a group, if the product of any two is another of the same collection. This may be illustrated outside the field of substitutions by the three cube roots of unity 1, — ^ -)- J y 3, — — V^^ the product of any two being another of the same collection. The six substitutions s„ = 1, Si ;= (xyz), s,= {x::y) , Ss = x(yx), s, = yt,zx), s,, = .2 (a-i/) also form a group. The number of .substitutions of a group is its order, and tliis is always a factor of »!, Thus in the group given tlie order is 6, and this is a factor of 3!. If all substitutions of a group H are contained in anotlier group G, H is called a sub-group of G and the order of H is a factor of that of G. A group whose operations are all permutable with one another is called an Abelian group. Lagrange (1770) was one of the first to under- take a scientific treatment of substitutions in connection with the theory of the quintic equa- tion. He invented a 'calcul des combinaisons,' the first real step toward the theory of substi- tutions. Euffini (1799) was the next to under- take a serious stud' of the subject, again in the attempt to show the impossibility of solving the quintic. To Galois (q.v. ), however, the honor of establishing the theory is usually ascribed. He found that if r^, r^, ^j, . . . r^ are the n roots of an equation, there is always a group of per- nuitations of the rs such that (1) every func- tion of the roots invariable by the substitutions of the group is rationally known, and ('2), re- ciprocally, every rationally determinable func- tion of the roots is invariable by the substitu- tions of the group, a discovery that eventually led to the proof of the insolubility of the quintic (Liouville's Journal, vol. xi.). Consult also (Euvres mathematiques de Galois (Paris, 1897). Cauchy was the first of the well- known French mathematicians to recognize the importance of the theory, and numerous im- portant propositions are due to him [Journal de Vieole polytechnique, 1815; Exercices d'analyse et de physique mathematique, vol. iii., Paris, 1844). Serret was the first to give a connected account of the theory (Cours d'algebre superi- eure. in the 3d ed.,' Paris, 1866). This wa» followed by Jordan's Traitc des substitutions et des equations algcbriques (Paris. 1870). Sylow (1872) was the first to treat the subject apart from its applications to equations (ilath. An- nalcn, vol, v.). He was followed by Netto,. whose Substitutionstheorie (Leipzig, 1882) was translated by Cole (Ann Arbor, 1892), thus making the theoiy accessible to English readers. Burnside, Theory of Groups of Finite Order (Cambridge, 1897), has brought the theory even more prominently before English and American scholars. The first to attempt to simplify the applications of the theory to the subject of equa- tions in an elementary text-book was Petersen, Thcorie der algebraischen Gleichungen (Copen- hagen, 1878). SUBSTITUTION, Theory of. See Chemis- try (historical section). SUBTRACTION (Lat. subtractio, a taking away, from .uihtralicre. to take away, from sub, under + trahere. to draw, drag). The inverse of addition, and one of the fundamental processes of arithmetic and algebra. It is the operation which has for its object, given the sum of two expressions and one of them, to find the other. The given sum is called the minuend, the given addend is called the subtrahend, and the addend
