EPODE EQUATION. year, when it is a common year. The epoch is one of the elements of a planet's orbit. E'PODE is the last part of the chorus of the ancient Greeks, which they sung after the strophe and antistrophe, when the singers had returned to their original place. The epode had its peculiar measure of syllables and number of verses. See CHORUS. EPPING, a town in the west of Essex county, England, in a pleasant healthy situation, at the north end of Epping Forest, 16 miles north-northeast of London. It has a very irregular appearance. Pop. (1861) 2102. It is noted for its cream, butter, sausages, and pork. It sends large quantities of butter to London. Epping Royal Forest, formerly under the name of Waltham Forest, where our ancient kings enjoyed much sport, covered all Essex, and extended almost to London. It is now limited to 60,000 acres in the south-west part of the county: Of this tract, only 12,000 acres are in wastes and woods, the rest being now enclosed as private property. In the forest, 5 miles from E., is Queen Elizabeth's hunting-lodge. Separated by the river Roding from Epping Forest is Hainault Forest, lately disforested. Here for many centuries a fair was held under the enormous Fairlop oak, not now existing, and a stag was yearly turned out in the Forest on Easter Monday, for the amusement of the public. To this day, a stag is turned out yearly for the amusement of Cockney huntsmen. EPROUVETTE is a machine for proving or testing the strength of gunpowder. It was invented or suggested in the last century by Robins, but was greatly improved by Dr Hutton. The gun eprouvette determines the strength of gunpowder by the amount of recoil produced. A small gun, usually a half-pounder,' is fixed to the lower end of an iron rod; its base being adjusted to an arm projecting from the rod: or else it is suspended from an iron frame. A horizontal steel axis is fixed to the rod or frame about which the gun may vibrate. A pointed iron rod or style projects downwards from the lower side of the gun, and touches a groove filled with soft wax; the groove is so shaped that, when the gun recoils, the point cuts a path for itself along this wax; and the length of this path determines the amount of recoil. Sometimes a brass graduated arc with an index is used instead of the pointed style and the waxed groove; but the principle of action is just the same. On the arc the recoil should vary from 26° for new fine-grain powder to 20° 5' for old powder of coarse grain. This system of proof is resorted to annually at minor and foreign stations for the proof of all powder in store, to ascertain the amount of deterioration; five rounds constitute the minimum proof. Before the eprouvette is resorted to, the powder must pass the test of specific gravity, by weighing not less than 55 lbs. to the cubic foot. powder, and is retained at its extreme state of propulsion by a ratchet wheel. is a small market-town on the margin of the E'PSOM (said to have originally been Ebbasham) Banstead Downs in Surrey, 15 miles south-southwest of London by road, and 14 miles by the London and South-Western Railway. The famed sulphate of magnesia springs of E. gave their name to the Epsom Salts formerly manufactured from them. This manufacture has been abandoned from the ease with which these salts can be made artificially. The Royal Medical College, erected on the Downs, and established in 1851, provides education for about 170 boys, the sons of medical men, and affords a home to decayed members of the profession and their widows. Pop. (1861) 4882. On the Downs, 11⁄2 mile south of the town, the famous E. horse-races are held yearly. They are said to have been instituted by Charles I., but have become of greater importance since the institution of the Derby Stakes in 1780 (see DERBY DAY). The races last four days, and as many as 100,000 persons often assemble to witness the most important of them. EPSOM SALT, or SU'LPHATE OF MAGwater of mineral springs, as at Epsom, Seidlitz, and NE'SIA (MgO+SO,HO), occurs not only in the many other places; but also as an efflorescence on the surface of various rocks, sometimes along with alum, as at Hurlet, in Renfrewshire; and on the ground, as in some parts of Spain and of the Russian steppes. It sometimes occurs snow-white and very pure, sometimes discoloured by impurities; and is crusts, flakes, granules, &c. Its crystals are prisms, either in the form of fine thread-like crystals, or in almost rectangular. For purposes of commerce, it is obtained by the action of dilute sulphuric acid upon magnesian limestone. See MAGNESIA. much in use in household medicine. Epsom salt is a well-known purgative remedy It may be given in doses from two drachms to one ounce, according to the effect required, in a tumbler of water. The disagreeable bitter taste is much relieved by, acidulating with nearly a teaspoonful of dilute sulphuric acid to each ounce of salt. E'PWORTH, a town in the north-west of Lincolnshire, England, 30 miles north-north-west of Lincoln. It chiefly consists of one street, above two miles long. The chief employments are hemp and flax dressing, rope-making and malting. Pop. (1861) 2197. John Wesley, founder of Methodism, as well as Kilham, founder of the seceding Wesleyans, was born here. E'QUABLE MOTION is that by which equal spaces are passed over in equal times. EQUALITY. See LIBERTY, EQUALITY, FRATERNITY. EQUATION, ANNUAL, one of the most conspicuous of the subordinate fluctuations in the EQUATION-EQUATION OF TIME. moon's motion, due to the action of the sun, which increases with its proximity to the earth and her on D. Tt + Then if A T + t + = D+ d and B 1 the equated time will = 12 A 2 satellite. It consists in an alternate increase and decreas in her longitude, corresponding with the earth's situation in its annual orbit, i. e., to its angular distance from the perihelion, and therefore (A-4B). When three or more debts are conhaving a year instead of a month, or aliquot part of cerned, the plan is to find by this formula the a month, for its period. For an explanation of the equated time for the first two, and then for their sum payable at their equated time, and the third, mode of its production, the reader is referred to and so on. The common rule is, however, suffiHerschel's Outlines of Astronomy, art. 738, et seq. The subject is too abstruse for explanation in this ciently correct for ordinary use. work. EQUATION, DIFFERENTIAL, is an equation involving differential coefficients (see CALCULUS); d3y dy + a x; from which it is required such is ᎾᏆ 03 to find the relation between and x. The theory of the solution of such equations is an extension of the integral calculus, and is a branch of study of the highest importance. EQUATION, FUNCTIONAL. See FUNCTIONS. EQUATION, LUNAR. See LUNAR THEORY. EQUATION OF EQUINOXES is the difference between the true position of the equinoxes, and the position calculated on the supposition that their motion is uniform. See PRECESSION. EQUATION OF LIGHT. In astronomical observations, the visual ray by which we see any body is not that which it emits at the moment we look at it, but that which it did emit some time before, viz., the time occupied by light in traversing the interval which separates it from us. If, then, the body be in motion, its aberration, as due to the earth's velocity, must be applied as a correction, not to the line joining the earth's place at the moment of observation with that occupied by the body, (as seen) at the same moment, but at that antecedent moment when the ray quitted it. Hence is derived a rule applied by astronomers for the rectification of observations made on a moving body, viz., from the known laws of its motion and the earth's, calculate its relative angular motion in the time taken by light to pass from it to the earth. This motion is the total amount of its apparent displacement. Its effect is to displace the body in a direction contrary to its apparent motion, an effect one part of which is due to aberration, properly so called (see ABERRATION), resulting from the composition of the motions of the earth and of light, and another part to the fact of the passage of light occupying time. The equation of light is the allowance to be made for the time occupied by the light in traversing a variable space. EQUATION OF PAYMENTS. The problem considered under this head in books of arithmetic is to find a time when, if a sum of money be paid by a debtor, which is equal to the sum of several debts payable by him at different times, no loss will be sustained by either the debtor or creditor. The rule generally given is as follows: Multiply each sum due by the time at which it is payable, and then divide the sum of the products by the sum of the debts: the quotient is the equated time. For example, if £10 be due at one month, and £20 at two months, find as an equivalent when the whole £30 10 × 1 + 20 × 2 Ans. may be paid at once. 13 30 months. This rule is, however, incorrect where the debts are unequal, because it takes no account of the balance of interest and discount. A correct rule for the case of two debts and simple interest is subjoined. Let d and D denote the debts, t and T the times of payment, and r one year's interest EQUATION OF THE CENTRE. If the earth moved uniforinly round the sun in a circle, it would be easy to calculate its longitude or distance from the line of equinoxes at any time. One year would be to the time since the vernal equinox as 360° to the arc of longitude passed over. But the orbit of the earth is not circular, nor is its motion uniform; the orbit is slightly elliptical, and the motion is quicker at perihelion than at aphelion. The true rule, then, for ascertaining the earth's longitude is contained in the following proportion: one year is to the time elapsed as the whole area of the earth's orbit is to the area swept over by the radius vector in the time. This is a deduction from Kepler's law (see CENTRAL FORCES), that, in planetary motion, equal areas (not angles) are swept over in equal times. The area swept over being ascertained from the laws of the earth's motion, and the elements of its orbit, it is a question of geometry to ascertain the angle corresponding to the area, or the calculated on the supposition that the earth moves true longitude. In astronomy, the longitude, as uniformly in a circle, is called the mean longitude of the earth; and it happens, from the orbit being, the mean and true longitude differ but slightly. as we said, but slightly different from a circle, that The quantity by which the true and mean longitudes differ is called the equation of the centre; and this is sometimes to be added to, and sometimes to be subtracted from the mean longitude, to obtain the true; and sometimes it is zero. EQUATION OF TIME. It will be seen from the article EQUATION OF THE CENTRE (q. v.) that the earth's motion in the ecliptic-or what is the same thing, the sun's apparent motion in longitude-is not uniform. This want of uniformity would of itself obviously cause an irregularity in the time of the sun's coming to the meridian on successive days; but besides this want of uniformity in the sun's apparent motion in the ecliptic, there is another cause of inequality in the time of its coming on the meridian-viz., the obliquity of the ecliptic to the equinoctial. Even if the sun moved in the equinoctial, there would be an inequality in this respect, owing to its want of uniform motion; and even if it moved uniformly in the ecliptic, there would be such an inequality, owing to the obliquity of its orbit to the equinoctial. These two independent causes conjointly produce the inequality in the time of its appearance on the meridian, the correction for which is the equation of time. When the sun's centre comes to the meridian, it is apparent noon, and if it moved uniformly on the equinoctial, this would always coincide with mean noon, or 12 o'clock on a good solar clock. But from the causes above explained, mean and apparent noon differ, the latter taking place sometimes as much as 164 minutes before the former, and at others as much as 14 minutes after. The difference for any day, called, as we have said, the equation of time, is to be found inserted in ephemerides for every day of the year. It is nothing or zero at four different times in the year, at which the whole mean and unequal motions exactly agra-viz., about the 15th of April, the 15th of June, the 31st August, and EQUATIONS. the 24th December. At all other times, the sun is either too fast or too slow for clock-time. In the ephemerides above referred to, the sign + or is prefixed to the equation of time, according as it is to be added to or subtracted from the apparent time to give the mean time. See NAUTICAL ALMANAC. (1), xy 11 EQUATIONS. An equation may be defined to be an algebraical sentence stating the equality of two algebraical expressions, or of an algebraical expression to zero. From another point of view, it is the algebraical expression of the conditions which connect known and unknown quantities. Thus 24, and (2), x2 + y2 = 52, are two equations expressing the relations between the unknown quantities x and y and known quantities. Generally, equations are formed from observations from which an object of inquiry may be inferred, but which do not directly touch the object. Thus, suppose we wish to ascertain the lengths of the sides of a rectangular board which we have no means of measuring, and that all the information we can get respecting it is, that it covers (say) 24 square feet, and that the square on its diagonal is (say) 52 square feet. From these facts, we can form equations from which we may determine the lengths of the sides. In the first place, we know that its area is equal to the product of its sides, and if we call these x and, = = 52. we have zy 24, the first of the equations above x 2 = kinds of equations, see EXPONENTIAL, FUNCTIONAL, and DIFFERENTIAL. The object of all computation is the determination of numerical values for unknown quantities, by means of the relations which they bear to other quantities already known. The solution of equa tions, accordingly, or, in other words, the evolution of the unknown quantities involved in them, is the chief business of algebra. But so difficult is this business, that, except in the simple cases where the unknown quantity rises to no higher than the second degree, all the resources of algebra are as yet inadequate to effect the solution of equations in general degree, or quadratic equations, as they are called, and definite terms. For equations of the second there is a rigorous method of solution by a general formula; but as yet no such formula has been discovered for equations even of the third degree. It is true, that for equations of the third and fourth degrees general methods exist, which furnish forof the roots. mulas which express under a finite form the values See CARDAN, and CUBIC EQUATIONS. But all such formulas are found to involve imaginary expressions, which, except in particular cases, make formulas are developed in infinite series, and the the actual computations impracticable till the imaginary terms disappear by mutually destroying instance (and all others are reducible to it), is in this one another. What is called Cardan's formula, for predicament whenever the values of the unknown quantity are all real; and accordingly, in nearly all But such cases, the values are not obtainable from the Equations are of several kinds. Simple equations are those which contain the unknown quantity in the first degree; thus, + 3 4, is a simple equation. Quadratic equations are those which contain the unknown quantity in the second degree x2+5x — 36 = 0, is a quadratic equation. Cubic and biquadratic equations involve the unknown in the third and fourth powers respectively. For the higher equations, there are no special names; they are said to be equations of the degree indicated by the highest power of the unknown which they contain. Simultaneous equations are those which involve two or more unknown quantities, and there must always be as many of them, in order to their determinate solution, as there are unknown quantities. The equations first mentioned-viz., 24 — x2 + y2 = 52, are simultaneous equations. It may be mentioned, that in the course of solving such equations the principal difficulties encountered are always ultimately the same as in the solution of equations containing only one unknown quantity. For instance, in the equations just given, if we substitute in the second the value of y as given by (24)2 the first, or y = we have x2 + 52, which x may be solved as a quadratic equation. The general theory of equations, then, is principally concerned with the solution of equations involving one unknown quantity only, for to this sort all others reduce themselves. Indeterminate equations are such as do not set forth sufficient relations between the unknown quantities for their absolute determination, and which accordingly admit of various solutions. (to which general form every equation of the nth = Chus. = 24 The rules for the solution of the simpler forms of equations are to be found in all elementary textbooks of algebra. It must suffice to notice here a few of the leading general properties of equations. By the roots of an equation are meant those values real or imaginary of the unknown which satisfy the equality; and it is a property of every equation to have as many roots and no more as there are units in its degree. Thus, a quadratic equation has two roots; a cubic equation, three; and a biquadratic, four. The quadratic equation x2+5x 360 has two roots, 9 and 4, which will be found to satisfy it. Further, the expression x2 + 5x 36 = (x-9)(x + 4) = 0; and generally if the roots of an equation 24 is an indeterminate equation, which degree can be reduced), are is satisfied by the values x = 3, y=8; or x = 6, y = 4; EQUATOR EQUIANGULAR. The factors, it will be observed, are formed thus: If+a, be a root, then x = a1, and a1 = 0 is the factor. If the root were - a1, then xa; and the factor would be x+a, = 0. Observing now the way in which, in multiplying a series of such factors, the coefficients of the resulting polynomial are formed, we arrive at this: that a complete equation cannot have a greater number of positive roots than these changes of sign from + to and from to in the series of terms forming its first member; and that it cannot have a greater number of negative roots than there are permanencies or repetitions of the same sign in proceeding from term to term. From the same source, many other general properties of equations, of value in their arithmetical solution, may be inferred. The subject is, however, too vast to be more than glanced at here. EQUATOR, CELESTIAL, is the great circle in the sky corresponding to the extension of the equator of the earth. EQUATOR, TERRESTRIAL, the great circle on the earth's surface dividing the earth into the northern and southern hemispheres, and half way between the poles. EQUATORIAL, an important astronomical - instrument, by which a celestial body may be observed at any point of its diurnal course. It consists of a telescope attached to a graduated circle, called the declination circle, whose axis penetrates at right angles that of another graduated circle called the hour circle, and is wholly supported by it. The pierced axis, which is called the principal axis of the instrument, turns on fixed supports; it is pointed to the pole of the heavens, and the hour circle is of course parallel to the equinoctial. In this position, it is easy to see that a great circle of the heavens corresponding to the declination circle, passes through the pole, and is an hour circle of the heavens. The telescope is capable of being moved in the plane of the declination circle. If, now, the instrument be so adjusted that the index of the declination circle must point to zero when an equatorial star is in the centre of the field of view of the telescope, and the index of the hour circle must point to zero when the telescope is in the meridian of the place, it is clear that when the telescope is directed to any star, the index of the declination circle will mark the declination of the star; and that on the other circle its right ascension. If the telescope be clamped when directed on a star, it is clear that, could the instrument be made to rotate on its principal axis with entire uniformity with the diurnal motion of the heavens, the star would always appear in the field of view. This motion of rotation is communicated to the instrument by clock-work. EQUESTRIAN ORDER, or E'QUITES. This body originally formed the cavalry of the Roman army, and is said to have been instituted by Romulus, who selected from the three principal Roman tribes 300 equites. This number was afterwards gradually increased to 3600, who were partly of patrician and partly of plebeian rank, and required to possess a certain amount of property. Each of these equites received a horse from the state; but about 403 B. C., a new body of equites began to make their appearance, who were obliged to furnish a horse at their own expense. These were probably wealthy novi homines, men of equestrian fortune, but not descended from the old equites (for it should be observed that the equestrian dignity was hereditary). Until 123 B. C., the equites were exclusively a military body; but in that year Caius Gracchus carried a measure, by which all the judices had to be selected from them. Now, for the first time, they became a distinct order or class in the state, and were called Ordo Equestris. In 70 B. C., Sulla deprived them of this privilege; but their power did not then decrease, as forming of the public revenues appears to have fallen into their hands. After the conspiracy of Catiline, the equestrian order, which on that memorable occasion had vigorously supported the Consul Cicero, began to be looked upon as a third estate in the Republic; and to the title of Senatus Populusque Romanus was added et Equestris Ordo. But, even in the beginning of the empire, the honour, like many others, was so indiscriminately and profusely conferred, that it fell into contempt, and the body gradually became extinct. As early as the later wars of the Republic, the equites had ceased to constitute the common soldiers of the Roman cavalry, and figure only as officers. the EQUESTRIAN STATUE, the representation of a man on horseback. Equestrian statues were awarded as a high honour to military commanders and persons of distinction in Rome, and latterly were, for the most part, restricted to the emperors, the most famous in existence being that of the Emperor Marcus Aurelius, which now stands in the Piazza of the Capitol at Rome. It is the only ancient equestrian statue in bronze that has been preserved; an exemption which it probably owed to the fact, that for centuries it was supposed to be a statue of Constantine. The action of the horse is so fine, and the air of motion so successfully given to it, that Michael Angelo is said to have called out to it 'Cammina!'--(Go on, then!). It was originally gilt, and traces of the gilding are still visible on the horse's head. So highly is this statue prized, not only for its artistic but its historical value, that an officer used regularly to be appointed by the Roman government to take care of it, under the designation of the Custode del Cavallo. On the occasion of the rejoicings by which Rienzi's elevation to the tribuneship was celebrated in 1347, wine was made to run out of one nostril and water out of the other of this famous horse. The statue then stood in front of the Church of St John Lateran, near to which it was found, and a bunch of flowers has always been presented annually to the chapter of that basilica, in acknowledgment of ownership, since it was removed to its present site on the Capitol. All European capitals are adorned, or disfigured, by numerous equestrian statues, London belonging pre-eminently to the latter category. EQUESTRIANISM. See HORSEMANSHIP. EQUIA NGULAR, having equal angles. A figure is said to be equiangular all whose angles are equal to one another, as a square, or any regular polygon. Also triangles and other figures are said to be equiangular one with another whose corres ponding angles are equal. EQUIDE-EQUISETUM EQUIDE, or SOLIDUNGULA (Lat. solidhoofed), a family of mammalia of the order Pachydermata, containing only a small number of species, which so nearly resemble each other that almost all naturalists agree in referring them to one genus, Equus. They are distinguished from all other quadrupeds by the complete consolidation of the bones of the toes, or the extraordinary development of one toe alone in each foot, with only one set of phalangeal and of metacarpal or metatarsal bones, and the extremity covered by a single undivided hoof. There are, however, two small protuberances (splint bones) on each side of the metacarpal or metatarsal bone (canon or cannon bone), which represent other toes. The E. have six incisors in each jaw, and six molars on each side in each jaw; the males have also two small canine teeth in the upper jaw, sometimes in both jaws, which are almost always wanting in the females. The molars of the E. have square crowns, and are marked by laminae of enamel with ridges forming four crescents. There is a wide space between the canine teeth and the molars. The stomach of the E. is simple, but the intestines are long, and the cæcum extremely large; the digestive organs being thus very different from those of the ruminants, but exhibiting an equally perfect adaptation to the same kind of not easily assimilated food. Another distinctive peculiarity of the E. is, that the females have two teats situated on the pubes, between the thighs. But notwithstanding these characters, so dissimilar to those of the ruminants, they approach them very much in their general conformation, and may be regarded as a connecting link between pachyderms and ruminants. The largely developed and flexible upper lip is a character which belongs to the former rather than to the latter order. The E. are now found in a truly wild state only in Asia and Africa. Fossil remains exist in the newer geological formations in great abundance in many parts of the Old World; very sparingly, however, in the New, although the bones of a peculiar and distinct species (Equus curvidens), belonging to the Pleiocene period, have been found in South America. The horse and the ass are by far the most important species of this family. The dziggethai has also been domesticated and made useful to man. Of the other species, the zebra, quagga, and dauw, it is generally believed that they are incapable of useful domestication. EQUILATERAL, having equal sides. A square is equilateral. The equilateral hyperbola is that whose axes and conjugate diameters are equal. EQUILIBRIUM, the state of rest or balance of a body or system, solid or fluid, acted upon by and nights are of equal length all over the world. ters, are names given to certain of the necessaries In those cases (in the receives a commission on the ground of meritorious service, an allowance of £100, if in the infantry, or £150, if in the cavalry, is made to him to provide an equipment. The equipment of a private soldier is often used as a name for the whole of his clothes, arms, and accoutrements collectively. The equipage of an army is of two kinds : it includes all the furniture of the camp, such as tents and utensils, under the name of camp-equipage; while field-equi page comprises saddle-horses, baggage-horses, and baggage-wagons. re EQUISE'TUM, a genus of Cryptogamous plants, the structure and affinities of which are not yet well understood, but which many botanists regard as constituting a sub-order of ferns, whilst others prefer to make it a distinct order, Equisetacea. The English name HORSE-TAIL is often given to all the species. They have a leafless, cylindrical, hollow, and jointed stem, each joint terminating in a membranous and toothed sheath, which encloses the base of the one above it. The fructification is at the summit of the stem in spikes, which somewhat sembles trobiles (cones), and are formed of scales their lower surface. The bearing spore-cases on spores are minute, oval, or round, green, and each accompanied with four elastic and hygrometrical threads. These threads are sometimes called elaters, but it is by no means certain that they are of the same nature with the E'QUINOXES. Sometimes the Equinoctial Points spiral filaments so called, (see EQUINOCTIAL) are called the equinoxes. More which are mixed with commonly, by the equinoxes are meant the times the spores of many when the sun enters those points, viz., 21st March Hepaticæ (q. v.). Each and 22d September, the former being called the thread terminates in a Vernal or Spring Equinox, and the latter the Autum- kind of club. The stems generally have lateral nal When in the equinoxes, the sun, through the branches, angular, but otherwise similar in structure earth's rotation on its axis, seems to describe the to the stem, growing in whorls from the joints; some ercle of the equator in the heavens, and the days | times the stem is simple; or fertile stenis are simple, various forces. See STATICS and HYDROSTATICS. EQUINO'CTIAL is the same with the celestial equator. See EQUATOR, CELESTIAL. The equinoctial points are those in which the equinoctial and the ecliptic intersect. See ECLIPTIC. Equinoctial time is time reckoned from the moment when the point of Aries passes the Vernal Equinox. See EQUINOXES. This instant is selected as a convenient central point of a uniform reckoning of time for the purposes of astronomical observers. Equisetum Telmatcia: summit o fertile stem, with fructification; 2, a scale, with its stalk (lateral view); 3, a spore, with its filaments unrolled; 4, a spore, with its filaments hygrometrically rolled up. |