Page:The New International Encyclopædia 1st ed. v. 07.djvu/16

* ELLIPSE. point (P), the sum of whose distances from two fixed points is constant. The two fixed points are called the foci (F and F') ; the diameter drawn through them is called the major axis, and the perpendicular bisector of this diameter the minor axis. It is also defined as the locus of a point which moves so that the ratio of its dis- tance from a fixed point, called the focus, to its ice from a fixed line (DD'), called the directrix, is constant and less than unity; the constant ratio is called the eccentricity of the ellipse and is equal to the ratio of the distance en the foci to the major axis. The limit of the ellipse, as the eccentricity approaches and the foci approach coincidence, is the circle. There are various contrivances for describing an el- ealled ellipsographs, or elliptic compasses. The simplest method is to fix the two ends of a thread to pins at the foci, and make a pencil move in the plane, keeping the thread taut. The end of the pencil will trace an el- lipse, whose major axis is equal to the length of t lie thread. Tlie Cartesian equation of the el- lipse placed symmetrically with respect to the axi- of coordinates (q.v.) is — + --- = 1, a, a- b- b being the semi-major and semi-minor axes re- Y spcctively. The eccentricity is e—^—- The polar equation of the ellipse is r=- — />, 1 -4-CCO80, where the parameter P = —. The centre O a bisects all chords; each diameter of a pair of conjug ' diameters bisects all chords I to tl iIni ; and the focal chord paral- lel to the directrix is called the latus rectum. Prom the equation of the ellipse, by means of the integral calculus, it- area i-- shown to be wab. I he lengl h of it ■ circumferem e is a pproxi- i ., I i ' ' l'8< l-'. r,/' : v-^-w'-^ ., ■•••;• There an pecial kinds of ellipses, for the liailie, and equal, liieb eiiNMlll 1 '. TO 'lie. 1807 I. ELLIP'SIS (], ;/, ,/,
 * in grammar and

i a of a word n< i e Bti iv i< or bi ntence in it a low itj and impre IPKOID (from Ok. I) I I Kind nl olid hounded by a nrface ol rder, the i ELLIS. spheroid being a special case. The latter lias a peculiar interest because the form of the earth is spheroidal. The Cartesian equation of an ellip- soid referred to its centre as origin, and its axes as axes of coordinates (q.v.) is— +7--I — =•= 1, a- b 2 c 2 where a, b, c are the semi-axes. By giving the value successively to x, y, s, the equations of three ellipses are obtained, which are the sec- tions of the ellipsoid, made by the respective co- ordinate planes. If any two of the quantities a, b, c are equal, one of the three sections is a circle, and the ellipsoid becomes an ellipsoid of revolution. The surface formed by the revolu- tion of an ellipse about its major axis is called a prolate spheroid, and that formed by the revo- lution about the minor axis is called an oblate spheroid, the latter being the general form of the earth. The term is often applied to the solid inclosed by the surface above defined. Ellipsoids play an important part in the theoiy of inertia. If on any line I through the origin O, a length OP be laid off, inversely proportional to the square root of the moment of inertia I of the line with respect to the given mass m, the locus of the point P will be a quadric surface called the ellipsoid of inertia or momental ellipsoid of the point O. The polar reciprocal (see Pole and Polar) of the momental ellipsoid with respect to a certain sphere is called the ellipsoid of gyration or the reciprocal ellipsoid. ELLIPTIC FUNCTIONS. See Functions. EL'LIS, Alexander John (1814-90). An English philologist and mathematician, born at Hoxton, Middlesex, and educated at Eton and Trinity College, Cambridge, lie was born under the surname Sharpe, but changed it to Ellis by royal license in 1825. He began the study of law in the Middle Temple, but gave it up for mathematics, first attracting attention by a translation of Ohm's Geist der mathematischen Analysis in 1847. It was as a phonologist and philologist, however, that Ellis was best known. lb' associated himself with Sir Isaac Pitman, with whom he formulated a system of printing which he culled 'phonotypy,' which added several new letters to the alphabet corresponding to sounds used in spoken language. His work formed the basis for all modern English phonetics. His studies in phonology led him naturally to a study of philology, and lie took high rank as an authority on both early English pronunciation and modern English dialects, Ilis greatest work in this Held was entitled On Early English P nunciation, with Special Reference to Bhakspere and Chaucer, which was published a1 intervals between 1869 and 1889. Another field in which Ellis achieved distinction was in the scientific theory of music, to the literature of which he contributed The Sensations of Tone as a Phy logical Basis for the Theory of Music 11875), based mi a German work by Helmholtz, and The History of Musical Pitch (1880). His other works include: liaise Taming (1842); Phonetics (1844) I Plea for Phoni tic Spelling (1848) ; Original fursery Rhymes for Boys and Oirls (1848); Algebra Identified icith Geometry ils7ti-. Practical Hints on the Quantitative Pronunciation 0/ Latin (1871); and Pronuncia- tion for Singi rs 1 1x77 ). ELLIS. Oeoboe Edward (1814-94). An American historian and editor, lie was born in