LEMNISCATE (from Gr. X wLo coc, ribbon)  , a quartic curve invented by Jacques Bernoulli (Acta Eruditorum, 1694) and afterwards investigated by Giulio Carlo Fagnano, who gave its principal properties and applied it to effect the division of a quadrant into 2.2m, 3.2n' and 5.2m equal parts . Following Archimedes, Fagnano desired the curve to be engraved on his tombstone . The complete analytical treatment was first given by Leonhard Euler . The lemniscate of Bernoulli may be defined as the locus of a point which moves so that the product of its distances from two fixed points is constant and is equal to the square of half the distance between these points . It is therefore a particular form of Cassini's oval (see Oval.) . Its cartesian equation, when the line joining the two fixed points is the axis of x and the middle point of this line is the origin, is (x2 + y2)2= 2a2(x2—y2) and the polar equation is r2=2a2 cos 20 . The curve (11g . 1) consists of two loops symmetrically placed about the coordinate axes . The pedal equation is r3=a2p, which shows that it is the first positive pedal of a rectangular hyperbola with regard to the centre . It is also the inverse of the same curve for the same point . It is the envelope of circles described on the central radii of an ellipse as diameters . The area of the complete curve is sae, and the length of any arc may be expressed in the form f(r—x°)—Idx, an elliptic integral sometimes termed the lemniscatic integral .

The name lemniscate is sometimes given to any cnanodal quartic curve having only one real finite

branch which is symmetric about the axis . Such curves are given by the equation x2—y2=ax4+ bx''y2+cy' . If a be greater than b the curve resembles fig . 2 and is sometimes termed the fishtail-lemniscate; if a be less than b, the curve resembles fig . 3 . The same name is also given to the first positive pedal of any central conic . When the conic is a rectangular hyper- bola, the curve is FIG . 4 . FIG . 5. the lemniscate of Bernoulli previously described . The elliptic lemniscate has for its equation (x2-+-y2)2=a2x2+b2y2 or r2=a2 cos2B+b2 sin 20 (a> b) . The centre is a conjugate point (or acnode) and the curve resembles fig .

4 . The hyperbolic lemniscate has for its equation (x2+y2)2=a2x2 —b2y2 or r2 =a'2 cos'O—b2 sin2 © . In this

case the centre is a crunode and the curve resembles fig . 5 . These curves are instances of Anicursal bicircular quartics .