An elementary treatise on cubic and quartic curves, by A. B. Basset.

THE LOGOCYCLIC CURVE.

83

136. If ( be the angle which the tangent at P makes with AP,
tan 4 = cos 0.
Taking the lower sign in (4) we have
tan ( = - tan APT= - rdO/dr = cos 0.
From the properties of inverse curves, it follows that ( is also
the angle which the tangent at P' makes with AP'.
137. If OK be drawn parallel to AP, and DT be drawn
perpendicular to AP meeting OK in T, the lines TP, TP' are the
tangents at P and P', and the locus of T is a cissoid.
Putting b= - a in the result of ~ 127, it follows that the locus
of T is the cissoid x (X2 + y2) = by2. Also since A 0= OX, it follows
that the line OK in ~ 128 is.parallel to AP. A direct proof may
of course be given.
138. The locus of the foot of the polar subtangent is a cardioid;
whilst that of the polar subnormal is a parabola.
Let ZAZ' be the polar subtangent; draw A Yperpendicular to
the tangent at P. Then
AZ = AP tan ( = b (sec 0- tan 0) cos 0
= b (1 + cos ZA 0).
Also if G, G' be the feet of the polar subnormals
AG = AP cot = b (sec 0 - tan 0) sec 0
b b
1 +sin0 + cosGAO'
139. Let the tangents from any point T on the asymptote touch
the curve in P and Q; and let the ordinates at these points meet
the curve again in P' and Q'; draw P'O, Q'O meeting the tangents
at Q and P in M and N respectively. Then the angles
PON= QOM = TAO.
Equation (9) of ~ 130 gives the vectorial angles referred to 0
as origin of the four tangents drawn from any point T. But if
T lie on the asymptote h = b = - a, and (9) becomes
3bz2- 2 z- b =........................(5),
whence
tan 09 + tan 02 = 2k/3b,
tan 01 tan 02 =-,
6-2