PROPERTIES OF THE PARABOLA. 245

line advance or recede in every possible position, its ex-tremity P, will trace the ellipsis ABCD.

On this principle, instruments have been constructedfor drawing an ellipsis by a continued motion, which aretrue elliptical compasses.

In a paper published in the Journal de VEcole Poly-technique, I have shown how this species of description,by a continued motion, may be applied to draw any ellip-soidal surface, using a right line, of which three ascer-tained points always remain in three fixed planes, while afourth, made to advance or recede in every direction, de-scribes an ellipsoidal surface. This method may be ap-plied in those drawings or operations which are requiredin constructing elliptical arches.

II. The Parabola, fig. 17, is formed from the coneABOZw, by a plane QR, parallel to one of the edges ofthe cone. It is a curve mnp, closed on one side, open onthe other, and extending into infinity, its two branchesnm, np, separating more and more.

The parabola MNP, fig. 18, has only one axis NL, inrelation to which its two branches, NM, MP, are symme-trical. It has one focus F.

Produce the axis by a quantity, NG = NF, the dis-tance of the focus from the summit of the parabola, anddraw through the point G, the right line XY, perpendi-cular to the axis. If we produce the reflected radius IKto H, on the line XY, the point I, of the parabola, willbe equally distant from the focus and from XY ; there-fore FI = HI. Take a square, with a cord fastened atF, and placed along X Y, having a second cord directedalong the square; if we hold the two cords in I, so thatFI = IH, and allow both the cords to be equally unroll-ed, as the square is removed from the axis, the point Iwill describe a parabola.

If we suppose the ellipsis gradually lengthened, the twofoci will gradually separate from each other. If we re-main at one of the foci, that part of the ellipsis which