involution solves such constructions as: given four tangents and one point, three tangents and two points, &c. If a tangent and its point of contact be given, it is only necessary to remember that a double point on the curve is given. A focus or directrix is equal to two conditions; hence such problems as: given a focus and three points; a focus, two points and one tangent; and a focus, one point and two tangents are soluble (very conveniently by employing the principle of reciprocation). Of practical importance are the following constructions:—(1) Given the axes; (2) given the major axis and the foci; (3) given the focus, eccentricity and directrix; (4) to construct an ellipse (approximately) by means of circular arcs.
(1) If the axes be given, we may avail ourselves of several constructions, (a) Let AA′, BB′ be the axes intersecting at right angles in a point C. Take a strip of paper or rule and mark off from a point P, distances Pa and Pb equal respectively to CA and CB. If now the strip be moved so that the point a is always on the minor axis, and the point b on the major axis, the point P describes the ellipse. This is known as the trammel construction.
(b) Let AA′, BB′ be the axes as before; describe on each as diameter a circle. Draw any number of radii of the two circles, and from the points of intersection with the major circle draw lines parallel to the minor axis, and from the points of intersection with the minor circle draw lines parallel to the major axis. The intersections of the lines drawn from corresponding points are points on the ellipse.
(2) If the major axis and foci be given, there is a convenient mechanical construction based on the property that the sum of the focal distances of any point is constant and equal to the major axis. Let AA′ be the axis and S, S′ the foci. Take a piece of thread of length AA′, and fix it at its extremities by means of pins at the foci. The thread is now stretched taut by a pencil, and the pencil moved; the curve traced out is the desired ellipse.
(3) If the directrix, focus and eccentricity be given, we may employ the general method for constructing a conic. Let S (fig. 2) be the focus, KX the directrix, X being the foot of the perpendicular from S to the directrix. Divide SX internally at A and externally at A′, so that the ratios SA/AX and SA′/A′X are each equal to the .jpg)
Fig. 2. eccentricity. Then A, A′ are the vertices of the curve. Take any point R on the directrix, and draw the lines RAM, RSN; draw SL so that the angle LSN = angle NSA′. Let P be the intersection of the line SL with the line RAM, then it can be readily shown that P is a point on the ellipse. For, draw through P a line parallel to AA′, intersecting the directrix in Q and the line RSN in T. Then since XS and QT are parallel and are intersected by the lines RK, RM, RN, we have SA/AX = TP/PQ = SP/PQ, since the angle PST = angle PTS. By varying the position of R other points can be found, and, since the curve is symmetrical about both the major and minor axes, it is obvious that any point may be reflected in both the axes, thus giving 3 additional points.
(4) If the axes be given, the curve can be approximately constructed by circular arcs in the following manner:—Let AA′, BB′ be the axes; determine D the intersection of lines through B and A parallel to the major and minor axes respectively. Bisect AD at E and join EB. Then the intersection of EB and DB′ determines a point P on the (true) curve. Bisect the chord PB at G, and draw through G a line perpendicular to PB, intersecting BB′ in O. An arc with centre O and radius OB forms part of a curve. Let this arc on the reverse side to P intersect a line through O parallel to the major axis in a point H. Then HA1 will cut the circular arc in J. Let JO intersect the major axis in O1. Then with centre O1 and radius OJ = OA1, describe an arc. By reflecting the two arcs thus described over the centre the ellipse is approximately described.
ELLIPSOID, a quadric surface whose sections are ellipses. Analytically, it has for its equation x2/a2 + y2/b2 + z2/c2 = 1, a, b, c being its axes; the name is also given to the solid contained by this surface (see Geometry: Analytical). The solids and surfaces of revolution of the ellipse are sometimes termed ellipsoids, but it is advisable to use the name spheroid (q.v.).
The ellipsoid appears in the mathematical investigation of physical properties of media in which the particular property varies in three directions within the media; such properties are the elasticity, giving rise to the strain ellipsoid, thermal expansion, ellipsoid of expansion, thermal conduction, refractive index (see Crystallography), &c. In mechanics, the ellipsoid of gyration or inertia is such that the perpendicular from the centre to a tangent plane is equal to the radius of gyration of the given body about the perpendicular as axis; the “momental ellipsoid,” also termed the “inverse ellipsoid of inertia” or Poinsot’s ellipsoid, has the perpendicular inversely proportional to the radius of gyration; the “equimomental ellipsoid” is such that its moments of inertia about all axes are the same as those of a given body. (See Mechanics.)
ELLIPTICITY, in astronomy, deviation from a circular or spherical form; applied to the elliptic orbits of heavenly bodies, or the spheroidal form of such bodies. (See also Compression.)
ELLIS (originally Sharpe), ALEXANDER JOHN (1814–1890), English philologist, mathematician, musician and writer on phonetics, was born at Hoxton on the 14th of June 1814. He was educated at Shrewsbury, Eton, and Trinity College, Cambridge, and took his degree in high mathematical honours. He was connected with many learned societies as member or president, and was governor of University College, London. He was the first in England to reduce the study of phonetics to a science. His most important work, to which the greater part of his life was devoted, is On Early English Pronunciation, with special reference to Shakespeare and Chaucer (1869–1889), in five parts, which he intended to supplement by a sixth, containing an abstract of the whole, an account of the views and criticisms of other inquirers in the same field, and a complete index, but ill-health prevented him from carrying out his intention. He had long been associated with Isaac Pitman in his attempts to reform English spelling, and published A Plea for Phonotypy and Phonography (1845) and A Plea for Phonetic Spelling (1848); and contributed the articles on “Phonetics” and “Speech-sounds” to the 9th edition of the Ency. Brit. He translated (with considerable additions) Helmholtz’s Sensations of Tone as a physiological Basis for the Theory of Music (2nd ed., 1885); and was the author of several smaller works on music, chiefly in connexion with his favourite subject phonetics. He died in London on the 28th of October 1890.
ELLIS, GEORGE (1753–1815), English author, was born in London in 1753. Educated at Westminster school and at Trinity College, Cambridge, he began his literary career by some satirical verses on Bath society published in 1777, and Poetical Tales, by “Sir Gregory Gander,” in 1778. He contributed to the Rolliad and the Probationary Odes political satires directed against Pitt’s administration. He was employed in diplomatic business at the Hague in 1784; and in 1797 he accompanied Lord Malmesbury to Lille as secretary to the embassy. On his return he was introduced to Pitt, and the episode of the Rolliad, which had not been forgotten, was explained. He found continued scope for his powers as a political caricaturist in the columns of the Anti-Jacobin, a weekly paper which he founded in connexion with George Canning and William Gifford. For some years before the Anti-Jacobin was started Ellis had been working in the congenial field of Early English literature, in which he was one of the first to arouse interest. The first edition of his Specimens of the Early English Poets appeared in 1790; and this was followed by Specimens of Early English Metrical Romances (1805). He also edited Gregory Lewis Way’s translation of select Fabliaux in 1796. Ellis was an intimate friend of Sir Walter Scott, who styled him “the first converser I ever saw,” and dedicated to him the fifth canto of Marmion. Some of the correspondence between them is to be found in Lockhart’s Life. He died on the 10th of April 1815. The monument erected to his memory in the parish church of Gunning Hill, Berks, bears a fine inscription by Canning.
ELLIS, SIR HENRY (1777–1869), English antiquary, was born in London on the 29th of November 1777. He was educated at Merchant Taylors’ school, and at St John’s College, Oxford, of which he was elected a fellow. After having held for a few months a sub-librarianship in the Bodleian, he was in 1800 appointed to a similar post in the British Museum. In 1827 he became chief librarian, and held that post until 1856, when he resigned on account of advancing age. In 1832 William IV. made him a knight of Hanover, and in the following year he received an English knighthood. He died on the 15th of January 1869. Sir Henry Ellis’s life was one of very considerable literary activity. His first work of importance was the preparation of a new edition of Brand’s Popular Antiquities, which appeared in 1813. In 1816 he was selected by the commissioners of public
