FLLIPSIS, in geometry, a curve line returning into itself, and produced from the section of a cone by a plane cutting both its sides, but not parallel to the base. See CONIC SECTIONS.

The easiest way of describing this curve, in plano, when the transverse and conjuares AB, ED, (Plate V. Miscell. fig. 1.) are given, is this: first take the points F, f, in the transverse axis A B, so that the distances CF, Cf, from the centre C, be each equal to AC-CD; or, that the lines FD, fD, be each equal to AC; then, hav. ing fixed two pins in the points F, f, which are called the foci of the ellipsis, take a thread equal in length to the transverse -axis A B; and fastening its two ends, one to the pin F, and the other to f, with another pin M stretch the thread tight; then if this pin M be moved round till it returns to the place from whence it first set out, keeping the thread always extended so as to form the triangle F.Mf, it will describe an ellipsis, whose axes are A B, D E.

The greater axis, A B, passing through the two foci Ff, is called the transverse axis; and the lesser one DE, is called the conjugate, or second axis: these two always bisect each other at right angles, and the eentre of the ellipsis is the point C, where they intersect. Any right line passing through the centre, and terminated by the curve of the ellipsis on each side, is called a diameter; and two diameters, which na turally bisect all the parallels to each other, bounded by the ellipsis, are called conjugate diameters. Any right line, not passing through the centre, but terminated by the ellipsis, and bisected by a diameter, is VOL. III.

called the ordinate, or ordinate-applicate' to that diameter; and a third proportional to two conjugate diameters, is called the latus rectum, or parameter of that diameter which is the first of the three proportionals.

The reason of the name is this: let BA, ED, be any two conjugate diameters of an ellipsis (fig. 2, where they are the two axes) at the end A, of the diameter A B, raise the perpendicular AF, equal to the latus rectum, or parameter, being a third proportional to AB, ED, and draw the right line BF; then if any point P be taken in BA, and an ordinate PM be drawn, cutting BF in N, the rectangle under the absciss A P, and the line PN will be equal to the square of the ordinate PM. Hence drawing NO parallel to AB, it appears that this rectangle, or the square of the ordinate, is less than that under the absciss AP, and the parameter AF, by the rectangle under AP and OF, or NO and OF; on account of which deficiency, Apollonius first gave this curve the name of an ellipsis, from ελλειπειν, to be deficient.

In every ellipsis, as A EBD, (fig. 2), the squares of the semi-ordinates MP, mp, are as the rectangles under the segments of the transverse axis APX PB, ApXp B, made by these ordinates respectively; which holds equally true of the circle, where the squares of the ordinates are equal to such rectangles, as being mean proportionals between the segments of the diameter. In the same manner, the ordinates to any diameter whatever, are as the rectangles under the segments of that diameter.

As to the other principal properties of

B

the ellipsis, they may be reduced to the following propositions. 1. If from any point M in an ellipsis, two right lines, MF, Mf, (fig. 1.) be drawn to the foci F, f, the sum of these two lines will be equal to the transverse axis A B. This is evident from the manner of describing an ellipsis. 2. The square of half the lesser axis is equal to the rectangle under the segments of the greater axis, contained between the foci and its vertices; that is, DC=AFX FB=Af xfB. 3. Every diameter is bisected in the centre C. 4. The transverse axis is the greatest, and the conjugate axis the least, of all diameters. 5. Two diameters, one of which is parallel to the tangent in the vertex of the other, are conjugate diameters; and vice versa, a right line drawn through the vertex of any diameter parallel to its conjugate diameter, touches the ellipsis in that vertex. 6. If four tangents be drawn through the vertices of two conjugate diameters, the parallelogram contained under them will be equal to the parallelogram contained under tangents drawn through the vertices of any other two conjugate diameters. 7. If a right line, touching an ellipsis, meet two conjugate diameters produced, the rectangle under the segments of the tangent, between the point of contact and these diameters, will be equal to the square of the semi-diameter, which is conjugate to that passing through the point of contact. 8. In every ellipsis, the sum of the squares of any two conjugate diameters is equal to the sum of the squares of the two axes. 9. In every ellipsis, the angles FGI, fGH, (fig. 1), made by the tangent HI, and the lines FG, G, drawn from the foci to the point of contact, are equal to each other. 10. The area of an ellipsis is to the area of a circumscribed circle, as the lesser axis is to the greater, and vice versa with respect to an inscribed circle; so that it is a mean proportional between two circles, having the transverse and conjugate axes for their diameters. This holds equally true of all the other corresponding parts belonging to an ellipsis.

The curve of any ellipsis may be obtained by the following series. Suppose the semi-transverse axis C B = r, the semi-conjugate axis CD=c, and the semi-ordinate a; then the length of the curve 4 r2 c2a-ra + 40 c

2 a3 MB = a++ 8 cr2 a2 + ra2 - 4cra 112 C12

&c. And, if

the species of the ellipsis be determined,

2.2.4.4.6.6 2.2.4.4.6.6.8.8, &c. is to 1, where d is equal to the difference of the squares of the axis applied to the square of the transverse axis.

ELLIPSIS, in grammar, a figure of syntax, wherein one or more words are not expressed; and from this deficiency it has got the name ellipsis.

ELLIPSIS, in rhetoric, a figure nearly allied to preterition, when the orator, through transport of passion, passes over many things: which, had he been cool, ought to have been mentioned. In preterition, the omission is designed; which, in the ellipsis, is owing to the vehemence of the speaker's passion, and his tongue not being able to keep pace with the emotion of his mind.

ELLIPTIC, or ELLIPTICAL, something belonging to an ellipsis. Thus we meet with elliptical compasses, elliptic conoid, elliptic space, elliptic stairs, &c. The elliptic space is the area contained within the curve of the ellipsis, which is to that of a circle described on the transverse axis, as the conjugate diameter is to the transverse axis; or it is a mean proportional between two circles, described on the conjugate and transverse axis.

ELLIPTOIDES, in geometry, a name used by some to denote infinite ellipses,

Of these there are several sorts: thus, if a y' = bx2 (a-x) it is a cubical elliptoid; and if a y = bx2 (a-x), it denotes a biquadratic elliptoid, which is an ellipsis of the third order in respect of the appollonian ellipsis.

ELLISIA, in botany, so called in memory of John Ellis, F. R. S. a genus of the Pentandria Monogynia class and order. Natural order of Luride. Borragineæ, Jussieu. Essential character: corolla fun