655 OF ENGINEERING FORMULÆ.
Curves—continued.
Equations of Curves.
The formula A x* + Bxy + Cyi + Das + F = 0, repn’-
gents an ellipse, parabola, or hyperbola, according as Bi - 4 A C
is positive in the ellipse ; is 0 in the parabola; or negative
in the hyperbola. _ , ...
The following are the equations of the principal curves
when x ~ the abscissa, y = the ordinate, a = the axis, p
the parameter.
Circle, y — /»Ja x - x1.
Ellipse, y = X' - (ax - æz).
a _________
Htperbola, y = A/ - (a x + æs).
Pababola, y — *Jpx. _______
/ p + x + \(2 yx + sca\
Catenary, y — p I hyp- log----------~ /
1st Conchoid, y = a2 ?>2 + 2®ä bx + a2 »«( when & _ (ii8.
2nd Conchoid, y = a* b4 — 2 a* b x + a x- )
tance of the generating point from the asymptote (or P B, see
diagram of Conchoid).
. / X '4
Cissoid, y — V --------- .
tl —■ 3/
CARi>roiDE,v/‘-6a^+2o;ii,2-6«æaj/ + a:'‘+12a2</2-%a»y
+ 3a2æ2=0. ____
/Jai — xi.
Lemniscate, y = x ............
sin. 0 (r — x~)
Quadbatbix, y- -- . ~ ; where 0 = the angle
V ri — (sin. 0),J
subtended by y, and r = the radius of the generating circle.
Spika l y — ---■ where c — th® circuniicreuce of the
’ c
circle in winch the spiral makes one revolution, r = the radius
of ditto, = x the distance of y from end of the spiral subtended
by the circumference of the generating circle, y = distance of
any part of the spiral measured from its centre, (bee Spiral.)