Page:EB1911 - Volume 09.djvu/752

 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 functions (1858); it is analogous to the before-mentioned trigonometrical solution of the cubic equation. The theory is reproduced and developed in Brioschi’s memoir, “Über die Auflösung der Gleichungen vom fünften Grade,” ''Math. Annalen'', t. xiii. (1877–1878).

26. The modern work, reproducing the theories of Galois, and exhibiting the theory of algebraic equations as a whole, is C. Jordan’s Traité des substitutions et des équations algébriques (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 substitutions defined analytically, and chiefly to linear substitutions; book iii. has four chapters, the first discussing the principles of the general theory, the other three containing 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, equations of Galois; the geometrical ones comprise Q. Hesse’s equation, R. F. A. Clebsch’s equations, lines on a quartic surface having a nodal line, singular points of E. E. Kummer’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 transcendents. And on this last subject, solution of equations by transcendents, we may quote the result—“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”; and again (but with a reference to a possible case of exception), “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.” (See also .)

—For the general theory see W. S. Burnside and A. W. Panton, The Theory of Equations (4th ed., 1899–1901); the Galoisian theory is treated in G. B. Matthews, Algebraic Equations (1907). See also the ''Ency. d. math. Wiss.'' vol. ii.

EQUATION OF THE CENTRE, in astronomy, the angular distance, measured around the centre of motion, by which a planet moving in an ellipse deviates from the mean position which it would occupy if it moved uniformly. Its amount is the correction which must be applied positively or negatively to the mean anomaly in order to obtain the true anomaly. It arises from the ellipticity of the orbit, is zero at pericentre and apocentre, and reaches its greatest amount nearly midway between these points. (See and .)

 EQUATION OF TIME, the difference between apparent time, determined by the meridian passage of the real sun, and mean time, determined by the passage of the mean sun. It goes through a double period in the course of a year. Its amount varies a fraction of a minute for the same date, from year to year and from one longitude to another, on the same day. The following table shows an average value for any date and for the Greenwich meridian for a number of years, from which the actual value will seldom deviate more than 20 seconds until after 1950. The + sign indicates that the real sun reaches the meridian after mean noon; the &minus; sign before mean noon.

Table of the Equation of Time.

 EQUATOR (Late Lat. aequator, from aequare, to make equal), in geography, that great circle of the earth, equidistant from the two poles, which divides the northern from the southern hemisphere and lies in a plane perpendicular to the axis of the earth; this is termed the “geographical” or “terrestrial equator.” In astronomy, the “celestial equator” is the name given to the great circle in which the plane of the terrestrial equator intersects the celestial sphere; it is consequently equidistant from the celestial poles. The “magnetic equator” is an imaginary line encircling the earth, along which the vertical component of the earth’s magnetic force is zero; it nearly coincides with the terrestrial equator.

 EQUERRY (from the Fr. écurie, a stable, through its older form escurie, from the Med. Lat. scuria, a word of Teutonic origin for a stable or shed, cf. Ger. Scheuer; the modern spelling has confused the word with the Lat. equus, a horse), a contracted form of “gentleman of the equerry,” an officer in charge of the stables of a royal household. At the British court, equerries are officers attached to the department of the master of the horse, the first of whom is called chief equerry (see, ).

 EQUIDAE, the family of perissodactyle ungulate mammals typified by the horse (Equus caballus); see. According to the older classification this family was taken to include only the forms with tall-crowned teeth, more or less closely allied to the typical genus Equus. There is, however, such an almost complete graduation from the former to earlier and more primitive mammals with short-crowned cheek-teeth, at one time included in the family Lophiodontidae (see ), that it has now become a very general practice to include the whole “phylum” in the family Equidae. The Equidae, in this extended sense, together with the extinct Palaeotheriidae, are indeed now regarded as forming one of four main groups into which the Perissodactyla are divided, the other groups being the Tapiroidea, Rhinocerotoidea and Titanotheriide. For the horse-group the name Hippoidea is employed. All four groups were closely connected in the Lower Eocene, so that exact definition is almost impossible.

In the Hippoidea there is generally the full series of 44 teeth, but the first premolar is often deciduous or wanting in the lower or in both jaws. The incisors are chisel-shaped, and the canines tend to become isolated so as in the now specialized forms to occupy nearly the middle of a longer or shorter gap between the incisors and premolars. In the upper molars the two outer columns of the primitive tubercular molar coalesce to form an outer wall, from which proceed two crescentic transverse crests; the connexion between the crests and the wall being imperfect or slight, and the crests themselves sometimes tubercular. Each of the lower molars carries two crescentic ridges. The number of toes ranges from four to one in the fore-foot, and from three to one in the hind-foot. The paroccipital, postglenoid and post-tympanic processes of the skull are large, and the latter always distinct. Normally there are no traces of horn-cores. The calcaneum lacks the facet for the fibula found in the Titanotheroidea.

In the earlier Equidae the teeth were short-crowned, with the premolars simpler than the molars; but there is a gradual tendency to an increase in the height of the crowns of the teeth, accompanied by increasing complexity of structure and the filling up of the hollows with cement. Similarly the gap on each side of the canine tooth in each jaw continues to increase in