About this book
My library
Books on Google Play
181. u is convergent or divergent according as the ratio of
Ux+1
to Ux1 when x = ∞, is less or greater than unity 240
182. Σaƒ(x) is convergent or divergent according as [ƒ (x) dx
is finite or infinite
241
244
183. Examples of the criterion given in the preceding article 243
184. On comparable and incomparable series
185-188. Tests of convergence and divergence of series derived
from the comparison of series with known series
245
189. Directions as to the application of the preceding tests
SECTION 2.-The series of Taylor and Maclaurin.
190. Criterion of convergence of these series
191. Proof of Taylor's series by definite integration, and the
value of its remainder as a definite integral
249
250
252
192. Proof of Maclaurin's series by definite integration, and the value of its remainder as a definite integral
SECTION 3.-The Development of Series by means of Single
Definite Integration.
193. The definite integral of a function expressed in a convergent
series ..
253
194. Examples of such definite integrals
254
195. Other series derived by definite integration from the pre-
ceding..
256
SECTION 4.-On Periodic Series, and on Fourier's Integral.
197. The mode of expanding a function in a periodic series
198. The geometrical interpretation of the discontinuity of the
199. The value of the function at the points of discontinuity 265
200. Forms of the functions when the limits cover the complete
201. Another form of the theorems given in the preceding
Article
267
202. Examples of these theorems
268
203. Discontinuous functions exhibited as periodic series
204. Fourier's theorem
CHAP. VIII.
ON MULTIPLE INTEGRATION, AND THE TRANSFORMATION OF
MULTIPLE INTEGRALS.
SECTION 1.-On Double, Triple, and Multiple Integration.
206. The extension of single definite integration to multiple in-
tegration
207. The problem of multiple integration further stated and
developed
208. The explanation of the symbolism
..
279
280
281
209. Conditions to be satisfied in multiple integration, as to the finiteness of the element-function and the order of in-
211. The general problem of transformation and its symbols
212. Another more general form of the same problem
213. Examples of the formula for transformation
214. Mode of assigning the new limits in the transformed in-
tegral ..
290
215. Examples of transformation of definite multiple integrals
216. Cases in which the limits of the transformed integral are
made constant by the transformation
293
296
SECTION 3.-The Differentiation of a Multiple Integral with respect
to a Variable Parameter.
217. The differentiation of a definite multiple integral in its most
general form
218. The differentiation of the same, when one definite integra-
tion has been effected ..
297
298
CHAP. IX.
QUADRATURE OF SURFACES, PLANE AND CURVED.
SECTION 1.-Quadrature of Plane Surfaces; Cartesian Coordinates.
222. Quadrature of a plane surface contained between two given
curves..
223. Examples in illustration
224. Quadrature determined by means of substitution
225. Quadrature determined when the axes are oblique
307
308
310
311
SECTION 2.-Quadrature of Plane Surfaces; Polar Coordinates.
316
230. Investigation of the surface-element in terms of r and Р
231. Quadrature of a surface between a curve, its evolute, and
two bounding radii of curvature
SECTION 3.-Quadrature of Surfaces of Revolution.
317
233. The area of a surface, when the generating plane curve
revolves about the axis of y
321
234. The area of a surface of revolution when the axis of revolu-
tion is parallel to the axis of x
322
235. The area of a surface of revolution when the generating
curve is referred to polar coordinates
323
239. The same expressed in terms of certain subsidiary angles..
240. M. Catalan's interpretation of the same
328
330
241. Certain theorems relating to the surface of the ellipsoid 332
242. The value of the surface-element in terms of polar co-
243. The same value deduced by means of transformation from
the value expressed in rectangular coordinates ..
333
244. Examples of quadrature of curved surfaces
335
SECTION 5.-Gauss' system of Curvilinear Coordinates.
245. Explanation of the system, and examples of the same
246. Geometrical explanation and interpretation
247. The value of a length-element in terms of the same..
248. The value of a surface-element in terms of the same, and
344
345
249. The geometrical interpretation of the general formula for
the transformation of a double integral
250. More general forms of curvilinear coordinates ..
CHAP. X.
CUBATURE OF SOLIDS,
SECTION 1.-Cubature of Solids of Revolution.
251. Investigation of the volume-element, when the axis of x is
the axis of revolution ..
252. Examples in illustration
253. Investigation of volume-element, when axis of y is that of
revolution ..
349
254. Investigation of the volume-element of a solid of revolution,
when the generating area is referred to polar coor-
dinates
350
352
255. A similar process of cubature extended to volumes gene-
rated by plane areas moving according to other laws
256. The cubature of a solid of revolution when the generating
area is referred to an axis parallel to that of revolution 353
SECTION 2.-Cubature of Solids bounded by any curved Surface.
257. Investigation of volume-element
258. Examples of cubature
354
356
259. A simplification when the element-function is of the form
260. The modification of the general form in its application to
solids bounded by cylinders
261. Other transformations of the integral expressing the volume
262. Cubature of a volume referred to polar coordinates ..
263. Examples of the process of the preceding article
362
364
CHAP. XI.
ON SOME QUESTIONS IN THE CALCULUS OF PROBABILITIES, AND ON THE
DETERMINATION OF MEAN VALUES.
SECTION 1.-On the Calculus of Probabilities.
264. Elementary principles and definitions of the Calculus of
Probabilities
d
366
265. The problem of probabilities for which infinite summation is required. Examples of the problem
367
266. Examples of similar problems for which integration is re-
quired..
370
267. On combination of possible events, and the curve of pos-
268. On combination of errors of observation; and the measure
of precision of observation ..
269. The curve of probability
374
376
379
270. The probability of a given error of observation determined 379
271. The probability of an event and of the precedent cause
272. The probability of a future event derived from past events
273. The probability of the action of two contradictory causes
274. The preponderance of probability of two contradictory
causes..
SECTION 2.-The Determination of Mean Values.
275. The definition and process of finding a mean value.
amples of mean value..
380
381
CHAP. XII.
REDUCTION OF MULTIPLE INTEGRALS.
SECTION 1.-Reduction of Multiple Integrals by simple application
of the Gamma-function.
276. Importance of reducing multiple integrals
389
277. Cauchy's method of reduction when the limits are constant
278. Example of the process
279. Reduction effected by means of the Gamma-function when
282. Liouville's theorem, when the limits are given by an in-
equality
397
SECTION 2.-Dirichlet's Method of Reduction by means of a
Factor of Discontinuity. Fourier's Integral.
286. The application of a more general factor of discontinuity..
405