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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

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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

equation .. .. .. .. .. .. .. .. ..

366. Similar interpretation of a partial differential equation .. 517

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

order and first degree requires n arbitrary constants 518

368. Definition of general integral, particular integral, singular

solution ..

520

369, 370. Integration by separation of the variables.. .. .. 520

SECTION 2.—Exact Total Diferential 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 524

373. Total differential equations of three variables ; criterion of

exactness and process of integration .. .. .. .. 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.. .. .. .. 529

378. Geometrical interpretation of homogeneous equations .. 531

379. The substitution required for separation of the variables

shewn to be equivalent to multiplication by an inte-

grating factor .. .. .. .. .. .. .. .. 531

380. An a posteriori proof that a homogeneous equation when

thus modified is an exact differential .. .. .. .. 532

381. Another form reducible to an homogeneous equation ..

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