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

xiv

CONTENTS.

CHAPTER X.
SPECIAL QUARTICS.
ART. PAGE
243. Object of Chapter........ 162
244. Definition of the cassinian and its equation... 162
245. Orthogonal systems of cassinians and rectangular hyperbolas 163
246. Singularities of the cassinian...... 163
247. Foci of the cassinian....... 164
248-252. Various properties of the cassinian.... 164
253. The lemniscate of Bernoulli...... 168
254-255. Properties of the lemniscate..... 169
256. Equation of the evolute of the lemniscate... 170
257. The pedal of a central conic with respect to its centre 171
258. The lemniscate of Gerono..... 171
259. The oval of Descartes....... 172
260. Conditions that the curve should be real. Four cases. 173
261-264. Discussion of Cases I and IV. Properties of conjugate
ovals. Proof of the existence of three real collinear foci. 174
265. Form of the oval of Descartes, and positions of the foci. 176
266. Degeneration of the curve into a limagon... 177
267. Discussion of Cases II and III..... 179
268-270. Properties of the oval of Descartes.... 181
271-272. Properties of a cartesian, and its points of inflexion. 182
273. Foci of the oval of Descartes..... 183
274-275. Union of single foci when the curve degenerates into a
limagon or a cardioid........ 184
276-277. Cartesians with one real and two imaginary collinear foci. 184
278. Definition of the lima9on....... 186
279. The elliptic limagon........ 186
280-281. The curve has a triple, a double, and a single focus. 187
282. Positions of the double tangent and the points of inflexion. 188
283. The reciprocal polar of a lima9on...... 188
284. The limagon is the pedal of a circle..... 189
285. Tangents to a limagon........ 189
286. Locus of intersection of tangents at extremities of a chord
through the external focus is a cissoid.... 190
287. Ditto, when the chord passes through the node, is a nodal
circular cubic......... 190
288-291. Various properties of an elliptic limagon... 191
292. The hyperbolic limagon........ 192
293-296. Various properties of the hyperbolic limagon... 194
297. The trisectrix. Trisection of an angle..... 196
298. The cardioid......... 196
299. The cardioid is a one-cusped epicycloid..... 197
300. Properties of the cardioid...... 197

301-302. The orthoptic locus consists of a circle and a limagon.
303-304. Loci connected with the cardioid..

199
201