**LEMNISCATE** (from Gr.

The name **lemniscate** is sometimes given to any crunodal quartic curve having only one real finite branch which is symmetric about the axis.

2 and is sometimes termed the fishtail-**lemniscate**; if a be less than b, the curve resembles fig.

The same name is also given to the first positive pedal of any central conic. When the conic is a rectangular hyperbola, the curve is the **lemniscate** of Bernoulli previously described.

The elliptic **lemniscate** has for its equation (x 2 +31 2) 2 =a 2 x 2 +b 2 y 2 or r 2 = a 2 cos 2 9 +b 2 sin 20 (a> b).

The hyperbolic **lemniscate** has for its equation (x2 +y2)2 = a2x2 - b 2 y 2 or r 2 =a 2 cos 2 0 - b 2 sin 2 B.

The Nachlass contains further researches on this subject, and also researches (unfortunately very fragmentary) on the **lemniscate-function**, &c., showing that Gauss was, even before 1800, in possession of many of the discoveries which have made the names of N.

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.