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

182

SPECIAL QUARTICS.

269. The tangent at P bisects the angle between the focal
distance FP and the tangent at P to the circle through F~PFl.
If PL be the tangent to the circle,
FPG = TPQ= TQP = GPL.
270. The locus of S the middle point of PQ is a linmaon.
The polar equation, when F is the origin, is given by (1),
whence
FS = (rl + re) = (a - m2c cos 0)/(1 -2).
All the preceding propositions hold good for either of the
other two foci, provided P and Q are points satisfying the focal
properties (9) and (10). The footnote* contains a list of some
recent memoirs on this curve.
271. If any chord cut a cartesian in four points, the sum of
its distances from any focus is constant.
The equation of a cartesian referred to any focus is
r2 + 2r (a+ b cos 0) + 32 = 0;
let the equation of any straight line be r (A cos 0 + B sin 0) = 1;
then if we eliminate 0 between these two equations, we shall
obtain a quartic equation for r, in which the coefficient of r3 is
equal to 4a.
272. A cartesian has eight points of inflexion, and since the
curvature at such points vanishes and changes sign, the radius of
curvature becomes infinite at a point of inflexion. Hence the
denominator of the expression for the radius of curvature, when
equated to zero, furnishes the equation of the curve which passes
through the points of inflexion; and in the case of a cartesian
the curve is a circular cubic, whose equation may be found from
that of the curve by equating the value of d2y/dx2 to zero.
The last two propositions are true for all cartesians.
* Genocchi, Noun. Ann. 1855; Mathesis, 1884.
Zeuthen, Ibid. 1864, p. 304.
Sylvester, Phil. Mag. vol. xxxi. 1866.
D'Ocagne, Comp. Rend. 1883, p. 1424.
Liguine, Bull. de Darboux, 1882; Interm. des Math. 1896, p. 238.