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

174

SPECIAL QUARTICS.

duced. Recollecting that OD = a, OE = a/m, we obtain the following results:Case I. a>c; oc> m>-1.
CaseII. c>a>O; a/c>m>-o.
Case III. O >a >-c; a/c > >-oo.
Case IV. -c>a>- o; -1 >m>-oo.
If a or m lie outside these limits, the curve is imaginary.
Cases I. and IV.
261. We shall now examine the geometrical meaning of these
conditions, and we shall find it convenient to begin with the two
ovals
r + nr = a...........................(3),
r- mrl = a...................... (4),
in which a> c and 1 > m > 0.
With F as a centre describe a circle whose radius FR = a. On
FR take two points P and Q such that
PR QR
FIP FQ
R'
F VF2
Q'
Then P will be a point on (3) and Q a point on (4); also F1R
bisects the angle PF1Q. Let
FF1R =q, PFR = a, FR F = R;
then FP= csin (0-a) FQ csin( +a)
sin (R + a)' sin (R-a)
iwhence TFP rFQ- c2 (sin2 6 - sin2 a)
sin2 R - sin2 a