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159. Application of the method to total minima
160. The sufficiency of the process
161. Examples of the process
162. A consideration of a case wherein the requisite conditions
are not fulfilled
SECTION 4.-Maxima and minima of functions of three and more
independent variables.
163. Conditions of such singular values of a function of three
independent variables...
164. The requisite conditions in the most general case
165. The method of least squares
166. Examples of the method of least squares
258
259
260
262
263
264
266
270
SECTION 5.-Maxima and minima of functions when all the variables
are not independent.
167. Investigation of the most general case of many variables..
168. Discussion of the case of two variables which are connected
170. Object of the Chapter is the discussion of an algebraical
expression in its synthetical form
171, 172. The continuity of algebraical expressions
173. Proof that every equation has a root.. ...
174. If a is a root of ƒ (x), ƒ(x) is divisible by x-a
175. Every equation has as many roots as it has dimensions
176. The roots of f'(x) are intermediate to those of ƒ (x)
177. If f(x) has m equal roots, f'(x) has m-1 roots equal to
180. The criteria of the number of impossible roots of an equation
181. Fourier's Theorem
292
293
182. Des Cartes' rule of signs
295
183. Newton's method for finding a number greater than the
greatest root of an equation
184. Method of approximating to the root of an equation
PART II.
GEOMETRICAL APPLICATIONS.
CHAPTER IX.
ON GEOMETRY.
SECTION 1.-The adjustment of the principles of Pure Geometry
and Infinitesimal Calculus.
185. The application to space of the law of continuity ..
186. The definitions of some geometrical terms founded on the
said law ...
297
298
187. Corroboration of the preceding modes of interpretation
from some known examples
301
SECTION 2.-On an extension of the symbols of direction.
SECTION 3.
-
On the generation of some plane curves of higher orders.
193-206. On the Cissoid, the Witch, the Conchoid, the Lem-
niscata, the Logarithmic Curve, the Catenary, the Tractory,
the Cycloid, the Companion to the Cycloid, the Epitrochoid
and its varieties
311
SECTION 4.-General properties of algebraical curves of the nth degree.
207. Various forms, and the number of terms of an algebraical
equation of the nth degree .
208. The homogeneous forms, in terms of three variables
209. The order of the resultant of many algebraical equations..
210. The number of the points of intersection of two curves of
the mth and nth degree; the theorems of Newton and
Cotes
320
322
324
325
211. The number of given points through which a curve of the
nth degree may pass
327
212. The number of these points which may be on a curve of the
mth degree
330
CHAPTER X.
PLANE CURVES REFERRED TO RECTANGULAR COORDINATES.
SECTION 1.-Tangents and normals.
213. The principle of application of infinitesimal calculus to the
properties of curves ...
331
214. Definition of, and equation to, a tangent
332
215. Modification of the equation in case the function is (a)
implicit, (B) homogeneous..
333
216. Equation to the tangent, when the equation to the curve is
homogeneous and of three variables
334
217. Equation to the normal. . . . .
218. Values of ds, and of sin 7, cos T, sin, cos
336
219. Discussion of the equations to the tangent and the normal,
and of lines and quantities connected with them ....
220. Explanation of the course of a curve at points where
222. General properties of the tangent of a curve of the nth
degree.....
344
223. The number of tangents which can be drawn to a curve from
229. The number of normals which can be drawn to a curve from
a given point. Evolutes.....
355
230. The normal is the longest or shortest line that can be drawn
to a curve through a given point
356
SECTION 2.—On asymptotes to plane curves referred to rectangular
coordinates.
231. On rectilinear asymptotes
357
232. Method of determining asymptotes by means of expansion
in descending powers of a
358
233. Examples of the method
359
234. Asymptotes are also tangents to a curve at an infinite dis-
tance
360
235. Examples of the method
361
236. On asymptotes out of the plane of reference..
363
237. Curvilinear asymptotes
238, 239. The equations of asymptotes are factors of the general
equation of the curve
364
SECTION 3.-Direction of curvature and points of inflexion.
240. Direct proof that a curve is convex or concave downwards
12y
dx2
according as is positive or negative...
367
241. Another proof of the same theorem by means of an expansion 368
242. Examples in illustration.
370
243. Interpretation of the preceding results by the infinitesimal
method
371
244. Criterion of points of inflexion when the equation of the
curve is an implicit function
373
245. If the curve is of the nth degree the number of points of
inflexion within a finite distance of the origin cannot
250. An explicit function is explained which well exhibits some
of the peculiarities of cusps
384
251. The number of double points of a curve of the nth degree.
252. The number of cusps of a curve of the nth degree
385
387
253. The relation of a curve to its Hessian
388
254. Double points and the Hessian when the equation of the
curve is homogeneous and of three variables..
390
255. Triple points; and an example ....
391
256. Quadruple points
257. Algebraical property of equations giving rise to multiple
points ...
SECTION 5.
-On tracing curves by means of their equations.
258. General hints as to the analysis of the equation
259. Rules for tracing curves, and method to be followed
260. Examples of equations analysed and of curves traced
392
393
398
CHAPTER XI.
PLANE CURVES REFERRED TO POLAR COORDINATES.
SECTION 1.-Extension of the modes of interpretation, and on the
equations to some polar curves.
251. Interpretation of r and when affected with negative signs 411
262-265. Equations to the Spiral of Archimedes, the Reciprocal
Spiral, the Lituus, and the Logarithmic Spiral
266-268. Equations in terms of r and p of the involute of the
circle, the circle, and the epicycloid
SECTION 2.-Tangents and normals to polar curves.
269. The values of ds, and of lines and quantities connected
with the tangent and the normal
412
414
270. Other values of p ....
271. The same quantities in terms of p and r
272. Examples in illustration of the preceding
SECTION 3.-Asymptotes to polar curves.
273. Means of determining rectilinear asymptotes
274. Examples in illustration..
275. Asymptotic circles...
SECTION 4.-Direction of curvature and points of inflexion.
415
417
418
420
421
422
276. The convexity or concavity of a curve towards the pole
423
SECTION 5.-Tracing polar curves.
277 General hints for analysing the equations and tracing the