these, are pointed out in a long and friendly letter (30th of January 1819) from Robert Southey to the author.
Elliott’s wife brought him some money, which was invested in his father’s share of the iron foundry. But the affairs of the firm were then in a desperate condition, and money difficulties hastened his father’s death. Elliott lost all his money, and when he was forty years old began business again in Sheffield on a small borrowed capital. He attributed his father’s pecuniary losses and his own to the operation of the corn laws. He took an active part in the Chartist agitation, but withdrew his support when the agitation for the repeal of the corn laws was removed from the Chartist programme. The fervour of his political convictions effected a change in the style and tenor of his verse. The Corn-Law Rhymes (3rd ed., 1831), inspired by a fierce hatred of injustice, are vigorous, simple and full of vivid description. In 1833–1835 he published The Splendid Village; Corn-Law Rhymes, and other Poems (3 vols.), which included “The Village Patriarch” (1829), “The Ranter,” an unsuccessful drama, “Keronah,” and other pieces. He contributed verses from time to time to Tait’s Magazine and to the Sheffield and Rotherham Independent. In the meantime he had been successful in business, but he remained the sturdy champion of the poor. In 1837 he again lost a great deal of money. This misfortune was also ascribed to the corn laws. He retired in 1841 with a small fortune and settled at Great Houghton, near Barnsley, where he died on the 1st of December 1849. In 1850 appeared two volumes of More Prose and Verse by the Corn-Law Rhymer. Elliott lives by his determined opposition to the “bread-tax,” as he called it, and his poems on the subject are saved from the common fate of political poetry by their transparent sincerity and passionate earnestness.
An article by Thomas Carlyle in the Edinburgh Review (July 1832) is the best criticism on Elliott. Carlyle was attracted by Elliott’s homely sincerity and genuine power, though he had small opinion of his political philosophy, and lamented his lack of humour and of the sense of proportion. He thought his poetry too imitative, detecting not only the truthful severity of Crabbe, but a “slight bravura dash of the fair tuneful Hemans.” His descriptions of his native county reveal close observation and a vivid perception of natural beauty.
See an obituary notice in the Gentleman’s Magazine (Feb. 1850). Two biographies were published in 1850, one by his son-in-law, John Watkins, and another by “January Searle” (G. S. Phillips). A new edition of his works by his son, Edwin Elliott, appeared in 1876.
ELLIPSE (adapted from Gr. ἔλλειψις, a deficiency, ἐλλείπειν, to fall behind), in mathematics, a conic section, having the form of a closed oval. It admits of several definitions framed according to the aspect from which the curve is considered. In solido, i.e. as a section of a cone or cylinder, it may be defined, after Menaechmus, as the perpendicular section of an “acute-angled” cone; or, after Apollonius of Perga, as the section of any cone by a plane at a less inclination to the base than a generator; or as an oblique section of a right cylinder. Definitions in plano are generally more useful; of these the most important are: (1) the ellipse is the conic section which has its eccentricity less than unity: this involves the notion of one directrix and one focus; (2) the ellipse is the locus of a point the sum of whose distances from two fixedpoints is constant: this involves the notion of two foci. Other geometrical definitions are: it is the oblique projection of a circle; the polar reciprocal of a circle for a point within it; and the conic which intersects the line at infinity in two imaginary points. Analytically it is defined by an equation of the second degree of which the highest terms represent two imaginary lines. The curve has important mechanical relations, in particular it is the orbit of a particle moving under the influence of a central force which varies inversely as the square of the distance of the particle; this is the gravitational law of force, and the curve consequently represents the orbits of the planets if only an individual planet and the sun be considered; the other planets, however, disturb this orbit (see Mechanics).
The relation of the ellipse to the other conic sections is treated in the articles Conic Section and Geometry; in this article a summary of the properties of the curve will be given.
To investigate the form of the curve use may be made of the definition: the ellipse is the locus of a point which moves so that the ratio of its distance from a fixed point (the focus) to its distance from a straight line (the directrix) is constant and is less than unity. This ratio is termed the eccentricity, and will be denoted by e. Let KX (fig. 1) be the directrix, S the focus, and X the foot of the perpendicular from S to KX. If .jpg)
Fig. 1. SX be divided at A so that SA/AX = e, then A is a point on the curve. SX may be also divided externally at A′, so that SA′/A′X = e, since e is less than unity; the points A and A′ are the vertices, and the line AA′ the major axis of the curve. It is obvious that the curve is symmetrical about AA′. If AA′ be bisected at C, and the line BCB′ be drawn perpendicular to AA′, then it is readily seen that the curve is symmetrical about this line also; since if we take S′ on AA′ so that S′A′ = SA, and a line K′X′ parallel to KX such that AX = A′X′, then the same curve will be described if we regard K′X′ and S′ as the given directrix and focus, the eccentricity remaining the same. If B and B′ be points on the curve, BB′ is the minor axis and C the centre of the curve.
Metrical relations between the axes, eccentricity, distance between the foci, and between these quantities and the co-ordinates of points on the curve (referred to the axes and the centre), and focal distances are readily obtained by the methods of geometrical conics or analytically. The semi-major axis is generally denoted by a, and the semi-minor axis by b, and we have the relation b2 = a2 (1 − e2). Also a2 = CS·CX, i.e. the square on the semi-major axis equals the rectangle contained by the distances of the focus and directrix from the centre; and 2a = SP + S′P, where P is any point on the curve, i.e. the sum of the focal distances of any point on the curve equals the major axis. The most important relation between the co-ordinates of a point on an ellipse is: if N be the foot of the perpendicular from a point P, then the square on PN bears a constant ratio to the product of the segments AN, NA′ of the major axis, this ratio being the square of the ratio of the minor to the major axis; symbolically PN2 = AN·NA′ (CB/CA)2. From this or otherwise it is readily deduced that the ordinates of an ellipse and of the circle described on the major axis are in the ratio of the minor to the major axis. This circle is termed the auxiliary circle.
Of the properties of a tangent it may be noticed that the tangent at any point is equally inclined to the focal distances of that point; that the feet of the perpendiculars from the foci on any tangent always lie on the auxiliary circle, and the product of these perpendiculars is constant, and equal to the product of the distances of a focus from the two vertices. From any point without the curve two, and only two, tangents can be drawn; if OP, OP′ be two tangents from O, and S, S′ the foci, then the angles OSP, OSP′ are equal and also SOP, S′OP′. If the tangents be at right angles, then the locus of the point is a circle having the same centre as the ellipse; this is named the director circle.
The middle points of a system of parallel chords is a straight line, and the tangent at the point where this line meets the curve is parallel to the chords. The straight line and the line through the centre parallel to the chords are named conjugate diameters; each bisects the chords parallel to the other. An important metrical property of conjugate diameters is the sum of their squares equals the sum of the squares of the major and minor axis.
In analytical geometry, the equation ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represents an ellipse when ab > h2; if the centre of the curve be the origin, the equation is a1x2 + 2h1xy + b1y2 = C1, and if in addition a pair of conjugate diameters are the axes, the equation is further simplified to Ax2 + By2 = C. The simplest form is x2/a2 + y2/b2 = 1, in which the centre is the origin and the major and minor axes the axes of co-ordinates. It is obvious that the co-ordinates of any point on an ellipse may be expressed in terms of a single parameter, the abscissa being a cos ϕ, and the ordinate b sin ϕ, since on eliminating ϕ between x = a cos ϕ and y = b sin ϕ we obtain the equation to the ellipse. The angle ϕ is termed the eccentric angle, and is geometrically represented as the angle between the axis of x (the major axis of the ellipse) and the radius of a point on the auxiliary circle which has the same abscissa as the point on the ellipse.
The equation to the tangent at Θ is x cos Θ/a + y sin Θ/b = 1, and to the normal ax/cos Θ − by/sin Θ = a2 − b2.
The area of the ellipse is πab, where a, b are the semi-axes; this result may be deduced by regarding the ellipse as the orthogonal projection of a circle, or by means of the calculus. The perimeter can only be expressed as a series, the analytical evaluation leading to an integral termed elliptic (see Function, ii. Complex). There are several approximation formulae:—S = π(a + b) makes the perimeter about 1/200th too small; s =π√(a2 + b2) about 1/200th too great; 2s = π(a + b) + π√(a2 + b2) is within 1/30,000 of the truth.
An ellipse can generally be described to satisfy any five conditions. If five points be given, Pascal’s theorem affords a solution; if five tangents, Brianchon’s theorem is employed. The principle of
