OVAL, or as the name imports, egg-shaped, is the name given ori ginally to such a form as the section of an egg presents, round, but not circular. In mathematics it has received some extension of meaning. Any curve, or isolated branch of a curve, which returns into itself, would he called an oval; perhaps even a figure of eight would receive the name.
The curve having for its equation y= aix(x—a) (a and 6 being positive, and a less than b) has an oval extending from x=0 to x= a : but there is no curve whatever from x= a to x=6, or from x= —b to x=0. If a be small, the dimensions of the oval are email: and when a = 0 the equation becomes y=xJ(x2-62) in"which the oval has become a point (the origin), and is a conjugate point [CunvE], an isolated point which is not on any continuous branch.
Some conjugate points have none but imaginary values of dy:dx, some have one or more finite values. Thus when y=„1: there Is a conjugate point at the origin, and dy:dx is then imaginary : but when there is also a conjugate point at the origin, but dy:dx is 0. The meaning seems to be (as far as we can judge from a
few instances) that when the oval during its diminution, has axes which preserve a finite ratio to one another, so that its tangents fall in all direction; the ultimate value of dy:dx is imaginary. But when one of the axes diminishes without limit as compared with the other, so that, except near the ends of that axis, the tangents tend to assume one direction, there is an ultimata value of dy:dx which defines that direction. If our surmise be correct, a double or triple value of dyulx at the conjugate point would indicate the evanescence of a star-shaped oval, or of one which tends to assume that form as it diminishes. But this, with other points relating to the singular values of algebraic functions, has yet to be fully considered.