A Treatise on Infinitesimal Calculus ...

by a more general inequality

289. Another form of definite integral determined by means

of Fourier's theorem

406

408

CALCULUS OF VARIATIONS.

292. Difference as to operations and symbols between the dif-

CHAPTER XIII.

EXPOSITION OF THE PRINCIPLES OF THE CALCULUS OF VARIATIONS.

290. The calculus of variations is a calculus of continuous func-

tions

291. The view of it by the light of a problem

ferential calculus and the calculus of variations..

293. Further differences and coincidences..

414

294. Our ignorance limits the calculus of variations to certain

297. The variation of ds = (dx2+dy2+dz2)§ . .

301. The variation of definite integrals, and modes of simplify-

..

302. Variation of F(x, dx, d2x,.. y, dy, d2y,..)

ing such variations

304. Geometrical interpretation of the result of Article 303

305. Modification of the result when the variations become dif-

307. Modification of the result, when an equation of condition is

309. Modification of the result, when an equation of condition is

310. Calculation of a variation of a variation

311. The variation of a double integral

312. The modification of the preceding when the limits are given

by an equation of condition

440

442

446

313. Simplification when the element-function contains partial

derived functions of only the first order

447

314. The calculus of variations considers a function of an infinite

number of variables

..

CHAP. XIV.

APPLICATIONS OF THE CALCULUS OF VARIATIONS TO PROBLEMS

OF MAXIMA AND MINIMA.

448

SECTION 1.-Critical Values of Definite Integrals, whose Element-

Functions involve Variables and their Differentials.

315. The problem stated, and the methods of the differential

calculus shewn to be insufficient..

316. Determination of conditions for maxima and minima

317. Modification of the result when derived-functions, and not

differentials, enter into the element-function

318. The number of arbitrary constants contained in the final

result..

319. Particular cases

320. Problems of relative maxima and minima

321. Determination of maxima and minima, when the element-

function of the definite integral contains three variables 455

322-328. Problems of absolute maxima and minima

329-333. Problems of relative maxima and minima

334. A problem solved on the principle of Art. 314
335. The condition of equicrescence unnecessary in the applica-

tion of this process

SECTION 2.-On Geodesic Lines.

336. The equations of a geodesic, and the proof that the osculat-

ing plane of a geodesic is a normal plane of the surface 475

337. Another equation of a geodesic..

338. The radius of absolute curvature of a geodesic ..

339. The radius of torsion of a geodesic

340. Geodesics in reference to lines of curvature

341. On geodesic curvature and its measure

342. Geodesic circles, geodesic parallels, geodesic conics ..

343. Examples of geodesics on a sphere and on a cylinder

344. Geodesies on the surface of an ellipsoid; Joachimsthal's

345, 346. Theorems concerning geodesics on an ellipsoid

486

347, 348. The equations of geodesics in terms of curvilinear co-

ordinates

488

SECTION 3.- Investigation of Critical Values of a Definite Integral,

whose Element-Function involves Derived-Functions.

SECTION 4.-Discriminating Conditions of Maxima and Minima.

352. General considerations as to the required conditions

353. Statement of the requisites, and Jacobi's mode of satisfying

them ..

495

354. Proof that 8Hŋ dx is an exact differential ..

355. The form of its integral

356. The integral can always be found

357, 358. Two particular cases wherein the criteria are applied
359. Application of the criterion to the general case
360. The criterion applied to a case of relative critical value ..

SECTION 5.-Investigation of the Critical Values of a Double

DIFFERENTIAL EQUATIONS,

OR

INTEGRATION OF DIFFERENTIAL FUNCTIONS OF TWO

AND MORE VARIABLES.

CHAPTER XV.

THE INTEGRATION OF DIFFERENTIAL EQUATIONS OF THE first ordER.

SECTION 1.-General Considerations on Differential Equations.

364. Meaning of the term "differential equation"; definitions of

order, degree

513

365. Geometrical interpretation of an integral of a differential

367. The complete integral of a differential equation of the nth

order and first degree requires n arbitrary constants

368. Definition of general integral, particular integral, singular

solution

369, 370. Integration by separation of the variables...

SECTION 2.-Exact Total Differential Equations.

371. Criterion of exactness or condition of integrability in the

case of two variables, and examples in illustration

522

372. Another mode of integration, when the integral is definite

373. Total differential equations of three variables; criterion of

exactness and process of integration ..

524

525

374. The definite integral of a total differential equation of three

variables

527

375. A differential equation of n independent variables: number

of conditions necessary for exactness..

528

SECTION 3.-Homogeneous Equations of two Variables.

377. Integration of homogeneous equations by separation of the

variables; and examples of the same..

378. Geometrical interpretation of homogeneous equations

379. The substitution required for separation of the variables

shewn to be equivalent to multiplication by an inte-

grating factor

380. An a posteriori proof that a homogeneous equation when

thus modified is an exact differential..

381. Another form reducible to an homogeneous equation

SECTION 4.-The First Linear Differential Equation.

382. The variables separated by means of a substitution, and

examples..

383. Bernoulli's equation

SECTION 5.-Partial Differential Equations of the First Order
and First Degree.

384. Method of integrating partial differential equations, and of

introducing an arbitrary functional symbol

534

535

536

387. Partial differential equations of any number of variables..

388. Examples of integration of the same

543

545

SECTION 6.-Integrating Factors of Differential Equations.

389. Every differential equation of the first degree has an inte-

grating factor

390. And the number of these integrating factors is infinite

391. Mode of determining these integrating factors..

392. Examples in which integrating factors are determined

393. Examples in which the integrating factor is a function of

one variable

394. Mode of integrating P dx + q dy =0, when Pa ± Qy = 0

395. Cases in which (Qy+Px)-1 is an integrating factor of

..

396. Integrating factor of the linear differential equation of the

first order ..

554

397. Integrating factors of equations of three variables, and ex-

amples in illustration..

556

398. Application of the method to homogeneous equations

399. Another method of integrating differential equations of

three variables ..

560

562

400. Geometrical interpretation of the criterion of integrability

401. A method of integration, when the condition of integrability

is not satisfied

563

567

SECTION 7.-Singular Solutions of Differential Equations.

402. The general value of the integral of a differential equation

403. Conditions which the general integral must satisfy

404. Only one general form of function satisfies a differential

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