Page:Encyclopædia Britannica, Ninth Edition, v. 8.djvu/529

Rh E Q U - E Q U 509 first link of a long chain of questions relating to the trans formation and classification of irrationals. Returning to the question of -solution by radicals, it will be readily understood that by the adjunction of a radical the group may be diminished ; for instance, in the case of the general cubic, where the group is that of the .six permutations, by the adjunction of the square root which enters into the solution, the group is reduced to abc, bca, cab ; that is, it becomes possible to express rationally, in terms of the coefficients and of the adjoint square root, any function such as a-b + b 2 c + c 2 a which is not altered by the cyclical substitution a into b, b into c, c into a. And hence, to determine whether an equation of a given form is solvable by radicals, the course of investigation is to inquire whether, by the successive adjunction of radicals, it is possible to reduce the original group of the equation so as to make it ultimately consist of a single permutation. The condition in order that an equation of a given prime order n may be solvable by radicals was in this way obtained in the first instance in the form (scarcely intelligible with out further explanation) that every function of the roots x v x. }. . . x, a invariable by the substitutions x at+b for x k , must be rationally known ; and then in the equivalent form that the resolvent equation of the order 1.2. .n-2 must have a rational root. In particular, the condition in order that a quintic equation may be solvable is that Lagrange s resolvent of the order 6 may have a rational factor, a result obtained from a direct investigation in a valuable memoir by E. Luther, Crelle, t. xxxiv. (1847). Among other results demonstrated or announced by Galois may be mentioned those relating to the modular equations in the theory of elliptic functions ; for the trans formations of the orders 5, 7, 11, the modular equations of the orders G, 8, 12 are depressible to the orders 5, 7, 11 respectively ; but for the transformation, n a prime number greater than 11, the depression is impossible. The general theory of Galois in regard to the solution of equations was completed, and some of the demonstra tions supplied by Betti (1852). See also Serret s Cours (T Algebra superieure, 2d ed., 1854; 4th ed. 1877-78, in course of publication. 33. Returning to quintic equations, Jerrard (1835) estab lished the theorem that the general quintic equation is by the extraction of only square and cubic roots reducible to the form x^ + ax + b^O, or what is the same thing, to a? 5 + x + b = 0. The actual reduction by means of Tschirn- hausen s theorem was effected by Hermite in connexion with his elliptic-function solution of the quintic equation (1858) in a very elegant manner. It was shown by Cockle and Harley (1858-59) in connexion with the Jerrardian form, and by Cayley (1861), that Lagrange s resolvent equation of the sixth order can be replaced by a more simple sextic equation occupying a like place in the theory. The theory of the modular equations, more particularly for the case n = 5, has been studied by Hermite, Kro- necker, and Brioschi. In the case n = 5, the modular equation of the order 6 depends, as already mentioned, on an equation of the order 5 ; and conversely the general quintic equation may be made to depend upon this modular equation of the order 6 ; that is, assuming the solution of this modular equation, we can solve (not by radicals) the general quintic equation; this is Hermite s solution of the general quintic equation by elliptic func tions (1858); it is analogous to the before-mentioned trigonometrical solution of the c .ibic equation. The theory is reproduced and developed in Brioschi s memoir. &quot;Ueber die Auflosung der Gleichungen vom fiinften Grade,&quot; Math. Annalen, t, xiti. (1877-78). 34. The great modern work, reproducing the theories of Galois, and exhibiting the theory of algebraic equations as a whole, is Jordan s Traite des Substitutions et des Equa tions Alyebriques, Paris, 1870. The work is divided into four books book i., preliminary, relating to the theory of congruences ; book ii. is in two chapters, the first relating to substitutions in general, the second to substitu tions defined analytically, and chiefly to linear substitu tions ; book iii. has four chapters, the first discussing the principles of the general theory, the other three contain ing applications to algebra, geometry, and the theory of transcendents; lastly, book iv., divided into seven chapters, contains a determination of the general types of equations solvable by radicals, and a complete system of classification of these types. A glance through the index will show the vast extent which the theory has assumed, and the form of general conclusions arrived at; thus, in book iii., the algebraical applications comprise Abelian equations, equa tions of Galois; the geometrical ones comprise Hesse s equa tion, Clebsch s equations, lines on a quartic surface having a nodal line, singular points of Rummer s surface, lines on a cubic surface, problems of contact ; the applications to the theory of transcendents comprise circular functions, elliptic functions (including division and the modular equation), hyperelliptic functions, solution of equations by transcen dents. And on this last subject, solution of equations by transcendents, we may quote the result, &quot;the solution of the general equation of an order superior to five cannot be made to depend upon that of the equations for the division of the circular or elliptic functions ; &quot; and again (but with a reference to a possible case of exception), &quot; the general equation cannot be solved by aid of the equations which give the division of the hyperelliptic functions into an odd number of parts.&quot; (A. CA.) EQUITES, an order of men in the commonwealth of Rome to which there is no exact parallel in modern times. Their origin goes back to the earliest period of Roman history. During the reign of the kings they appear to have been of noble birth, the younger branches of patrician families. This we may infer from the statement of Polybius (vi. 20), that the knights noiv are chosen according to fortune, evidently intimating that their selection had previously depended on a different principle. Romulus is said to have divided them into three centuries or &quot; hundreds,&quot; each century being chosen from one of the three old Roman tribes, the Ramnes, Tities, and Luceres. Both Tullus Hostilius and Tarquinius added to their number ; but, according to Livy, it was Servius Tullius (576 B.C.) who first organized them into a distinct body, and compelled the state to contribute annually to their maintenance. It is difficult to perceive in what way we are to explain the statement of Livy (i. 43), that ten thousand pounds of brass were given to each for the purchase of a horse, an enormous sum when compared with that at which oxen and sheep Avere rated in the table of penalties. Every eques, of course, was bound to be provided with a good horse, and he may have been obliged to replace it if lost through any casualty in war. Its accoutrements, too, and a slave to take charge of it, were possibly all included in the sum. But whether, when the censor ordered the knight to sell his horse, it was the intention that the outfit money should be refunded to the state, we have no means of determining. Livy tells us also that the (fs hordearium or barley-money sup plied to each knight for the maintenance of his horse was obtained by a tax on widows and orphans. This certainly sounds strange, for it seems inconceivable that there should have been such a large number of rich widows ; and even though the word vidua is explained to mean every single woman, maiden as well as widow, the difficulty still remains. Beyond the hordearium the knights received no pay.