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Encyclopedia Britannica

LEMNISCATE (from Gr. X wLo coc, ribbon)

This article appears in Volume V16, Page 412 of the Encyclopedia Britannica.

Encyclopedia Britannica - Main :: LAP-LEO

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

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

COMPLETE

analytical treatment was first given by Leonhard

Euler

EULER, LEONHARD (1707-1783)

. The

lemniscate

LEMNISCATE (from Gr. X wLo coc, ribbon)

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

CONSTANT, BENJAMIN (1845-1902)

and is equal to the square of half the distance between these points. It is therefore a particular form of Cassini's

oval

OVAL (Lat. ovum, egg)

(see

Oval

OVAL (Lat. ovum, egg)

.). Its cartesian equation, when the line joining the two fixed points is the

axis

AXIS (Lat. for " axle ")

of x and the

middle

MIDDLE

point of this line is the origin, is (x2 + y2)2= 2a2(x2y2) 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

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

COMPLETE

curve is sae, and the length of any arc may be expressed in the form f(rx)Idx, an elliptic integral sometimes termed the lemniscatic integral.
The name

lemniscate

LEMNISCATE (from Gr. X wLo coc, ribbon)

is sometimes given to any cnanodal quartic curve having only one real finite branch which is symmetric about the

axis

AXIS (Lat. for " axle ")

. Such curves are given by the equation x2y2=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'Ob2 sin2 . In this

case

CASE
CASE, JOHN (d. 1600)

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

End of Article: LEMNISCATE (from Gr. X wLo coc, ribbon)

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