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

166 SPECIAL QUARTICS.
Also = 0 + CPH = '+ SPG 0' + r -,
= = 0/+ —7 +- sb',
+ ' = + ',
whence + = 0'- 0 + 7r,
dO' dO dy ddk
and therefore dO- dO d + di
ds ds ds ds
1 I I 1
or 1+
PL PM = PN p

251. The equation of the curve in terms of p and r is
rp (a2 - C2) + a2 = 2 (r 4 a4).
Let CP =r, p the perpendicular from C on to the tangent
at P. Draw SB, HB' perpendicular to CP; then
1 +1 sin b sin '
PL PM SP ~HP
HP sin + SP sinb'
a2_ C2
PB' + PB
a2_ C2
2r
a2 - C2)

and

1 _p
PN- r2 '

whence, by ~ 250,

2r _p l dp
a2 - c2 r2 r dr

r4
Integrating rp = 2 (a - ) + A.
To find the constant, we observe that at each of the vertices
r and p are each equal to a, whence A = (a4 - 2a2c2)/(a2- c2) and
rp (2 - c2) + o2c2 = I (r4 + a4).
This shows that the radius of curvature may be expressed in
the form
2 (a2 - c2) r3
P 3r4- a4 + 2a2c2'