LEMNISCATE (from Gr.
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.
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.