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-

209. Conditions to be satisfied in multiple integration, as to the

finiteness of the element-function and the order of in-

tegration

..

- SECTION 2.-The Transformation of Multiple Integrals.

282

283

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

284

286

287

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

218. The differentiation of the same, when one definite integra-

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.

SECTION 2.-Quadrature of Plane Surfaces; Polar Coordinates.

226. The differential expression of a surface-element

227. Examples illustrative of it..

312

313

228. The order of integrations inverted

315

229. Cases of various and curved limits

315

230. Investigation of the surface-element in terms of r and p

231. Quadrature of a surface between a curve, its evolute, and

two bounding radii of curvature

316

..

317

SECTION 3.-Quadrature of Surfaces of Revolution.

232. Investigation of the surface-element..

319

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

239. The same expressed in terms of certain subsidiary angles ..

240. M. Catalan's interpretation of the same

241. Certain theorems relating to the surface of the ellipsoid ..

242. The value of the surface-element in terms of polar co-

332

ordinates in space

333

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

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..

376

269. The curve of probability

379

379

380

270. The probability of a given error of observation determined

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. Ex-

amples of mean value..

381

382

383

384

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.

283. Dirichlet's factor of discontinuity

286. The application of a more general factor of discontinuity..