A Treatise on Electricity and Magnetism, Vol. 1 (1881) — part 2 of 28

211. Thomson's water-dropping machine. . „ 297

212. Holtz's electrical machine 298

213. Theory of regenerators applied to electrical machines . . . . 298

214. On electrometers and electroscopes. Indicating instruments

and null methods. Difference between registration and mea- surement 300

215. Coulomb's Torsion Balance for measuring charges 301

XXVi CONTENTS.

Art. Page

216. Electrometers for measuring potentials. Snow-Harris's and

Thomson's 304

217. Principle of the guard-ring. Thomson's Absolute Electrometer 305

218. Heterostatic method 308

219. Self-acting electrometers. — Thomson's Quadrant Electrometer 309

220. Measurement of the electric potential of a small body . . . . 312

221. Measurement of the potential at a point in the air 313

222. Measurement of the potential of a conductor without touching it 314

223. Measurement of the 'superficial density of electrification. The

proof plane 315

224. A hemisphere used as a test 316

225. A circular disk 317

226. On electric accumulators. The Ley den jar 319

227. Accumulators of measurable capacity 320

228. The guard-ring accumulator 321

229. Comparison of the capacities of accumulators 323

PAET II.

ELECTKOKINEMATICS. CHAPTER I.

THE ELECTBIC CUEEENT.

230. Current produced when conductors are discharged. 326

231. Transference of electrification 326

232. Description of the voltaic battery 327

233. Electromotive force 328

234. Production of a steady current 328

235. Properties of the current 329

236. Electrolytic action 329

237. Explanation of terms connected with electrolysis 330

238. Different modes of passage of the current 330

239. Magnetic action of the current 331

240. The Galvanometer , .... 332

CONTENTS. xxvii

CHAPTER II.

CONDUCTION AND BESISTANCE.

Art. Page

241. Ohm's Law 333

242. Generation of heat by the current. Joule's Law 334

243. Analogy between the conduction of electricity and that of heat 335

244. Differences between the two classes of phenomena 335

245. Faraday's doctrine of the impossibility of an absolute charge. . 336

CHAPTER III.

ELECTROMOTIVE FORCE BETWEEN BODIES IN CONTACT.

246. Volta's law of the contact force between different metals at the

same temperature 337

247. Effect of electrolytes . . , 338

248. Thomson's voltaic current in which gravity performs the part

of chemical action 338

249. Peltier's phenomenon. Deduction of the thermoelectric elec-

tromotive force at a junction 338

250. Seebeck's discovery of thermoelectric currents 340

251. Magnus's law of a circuit of one metal . . 340

252. Cumming's discovery of thermoelectric inversions 342

253. Thomson's deductions from these facts, and discovery of the

reversible thermal effects of electric currents in copper and

in iron 342

254. Tait's law of the electromotive force of a thermoelectric pair. . 343

CHAPTER IV.

ELECTROLYSIS.

255. Faraday's law of electrochemical equivalents 345

256. Clausius's theory of molecular agitation 347

257. Electrolytic polarization 347

258. Test of an electrolyte by polarization 348

259. Difficulties in the theory of electrolysis 348

260. Molecular charges .. .. .. .. 349

261. Secondary actions observed at the electrodes 351

262. Conservation of energy in electrolysis 353

263. Measurement of chemical affinity as an electromotive force . . 354

xxviii CONTENTS.

CHAPTER V.

ELECTROLYTIC POLARIZATION.

Art. Page

264. Difficulties of applying Ohm's law to electrolytes 356

265. Ohm's law nevertheless applicable 356

266. The effect of polarization distinguished from that of resistance 356

267. Polarization due to the presence of the ions at the electrodes.

The ions not in a free state 357

268. Relation between the electromotive force of polarization and

the state of the ions at the electrodes 358

269. Dissipation of the ions and loss of polarization 359

270. Limit of polarization 359

271. Ritter's secondary pile compared with the Leyden jar . . . . 360

272. Constant voltaic elements. — Daniell's cell . 363

CHAPTER VI.

MATHEMATICAL THEORY OF THE DISTRIBUTION OF ELECTRIC CURRENTS.

273. Linear conductors 367

274. Ohm's Law 367

275. Linear conductors in series 367

276. Linear conductors in multiple arc 368

277. Resistance of conductors of uniform section 369

278. Dimensions of the quantities involved in Ohm's law . . . . 370

279. Specific resistance and conductivity in electromagnetic measure 371

280. Linear systems of conductors in general 371

281. Reciprocal property of any two conductors of the system . . 373 282 a, 1. Conjugate conductors 373, 374

282. Heat generated in the system 374

283. The heat is a minimum when the current is distributed ac-

cording to Ohm's law 375

CHAPTER VII.

CONDUCTION IN THREE DIMENSIONS.

285. Notation 376

286. Composition and resolution of electric currents 376

287. Determination of the quantity which flows through any surface 377

288. Equation of a surface of flow .. 378

CONTENTS. xxix

Art. Pa*e

289. Eelation between any three systems of surfaces of flow . . . . 378

290. Tubes of flow 378

291. Expression for the components of the flow in terms of surfaces

of flow 379

292. Simplification of this expression by a proper choice of para-

meters 379

293. Unit tubes of flow used as a complete method of determining

the current 379

294. Current-sheets and current-functions .. 380

295. Equation of * continuity' 380

296. Quantity of electricity which flows through a given surface . . 382

CHAPTER VIII.

EESISTANCE AND CONDUCTIVITY IN THEEB DIMENSIONS.

297. Equations of resistance 383

298. Equations of conduction 384

299. Eate of generation of heat 384

300. Conditions of stability 385

301. Equation of continuity in a homogeneous medium 386

302. Solution of the equation 386

303. Theory of the coefficient I7. It probably does not exist .. 387

304. Generalized form of Thomson's theorem 388

305. Proof without symbols 389

306. Strutt's method applied to a wire of variable section. — Lower

limit of the value of the resistance 390

307. Higher limit 393

308. Lower limit for the correction for the ends of the wire .. .. 395

309. Higher limit 396

CHAPTER IX.

CONDUCTION THBOUGH HETEROGENEOUS MEDIA.

310. Surface-conditions 398

311. Spherical surface 400

312. §pherical shell 401

313. Spherical shell placed in a field of uniform flow 402

314. Medium in which small spheres are uniformly disseminated .. 403

315. Images in a plane surface .. .. 404

316. Method of inversion not applicable in three dimensions .. .. 405

317. Case of conduction through a stratum bounded by parallel

planes 405

xxx CONTENTS.

Art- Page

318. Infinite series of images. Application to magnetic induction .. 406

319. On stratified conductors. Coefficients of conductivity of a con-

ductor consisting of alternate strata of two different substances 407

320. If neither of the substances has the rotatory property denoted

by T the compound conductor is free from it 408

321. If the substances are isotropic the direction of greatest resist-

ance is normal to the strata .. .. e 408

322. Medium containing parallelepipeds of another medium .. .. 409

323. The rotatory property cannot be introduced by means of con-

ducting channels 410

324. Construction of an artificial solid having given coefficients of

longitudinal and transverse conductivity 411

CHAPTER X.

CONDUCTION IN DIELECTEICS.

325. In a strictly homogeneous medium there can be no internal

charge 412

326. Theory of a condenser in which the dielectric is not a perfect

insulator 413

327. No residual charge due to simple conduction 414

328. Theory of a composite accumulator 414

329. Kesidual charge and electrical absorption 416

330. Total discharge 418

331. Comparison with the conduction of heat 419

332. Theory of telegraph cables and comparison of the equations

with those of the conduction of heat 421

333. Opinion of Ohm on this subject 422

334. Mechanical illustration of the properties of a dielectric .. .. 423

CHAPTER XI.

MEASUREMENT OF THE ELECTRIC RESISTANCE OF CONDUCTORS.

335. Advantage of using material standards of resistance in electrical

measurements 426

336. Different standards which have been used and different systems

which have been proposed .. 426

337. The electromagnetic system of units 427

338. Weber's unit, and the British Association unit or Ohm .. .. 427

339. Professed value of the Ohm 10,000,000 metres per second .. 427

CONTENTS. xxxi

Art. Page

340. Reproduction of standards 428

341. Forms of resistance coils 429

342. Coils of great resistance 430

343. Arrangement of coils in series 430

344. Arrangement in multiple arc 431

345. On the comparison of resistances. (1) Ohm's method .. .. 432

346. (2) By the differential galvanometer 432

347. (3) By Wheatstone's Bridge 436

348. Estimation of limits of error in the determination 437

349. Best arrangement of the conductors to be compared .. .. 438

350. On the use of "Wheatstone's Bridge 440

351. Thomson's method for small resistances 442

352. Matthiessen and Hockin's method for small resistances .. .. 444

353. Comparison of great resistances by the electrometer .. .. 446

354. By accumulation in a condenser 447

355. Direct electrostatic method 447

356. Thomson's method for the resistance of a galvanometer .. .. 448

357. Mance's method of determining the resistance of a battery ,. 449

358. Comparison of electromotive forces 452

CHAPTER XII.

ELECTEIC RESISTANCE OF SUBSTANCES.

359. Metals, electrolytes, and dielectrics 454

360. Resistance of metals 455

361. Resistance of mercury .. .. 456

362. Table of resistance of metals 457

363. Resistance of electrolytes 458

364. Experiments of Paalzow 458

365. Experiments of Kohlrausch and Nippoldt 459

366. Resistance of dielectrics .. 460

367. Gutta-percha 462

368. Glass 462

369. Gases 463

370. Experiments of Wiedemann and Ruhlmann 463

ELECTEICITY AND MAGNETISM,

PRELIMINARY.

ON THE MEASUREMENT OF QUANTITIES.

1.] EVERY expression of a Quantity consists of two factors or components. One of these is the name of a certain known quan- tity of the same kind as the quantity to be expressed, which is taken as a standard of reference. The other component is the number of times the standard is to be taken in order to make up the required quantity. The standard quantity is technically called the Unit, and the number is called the Numerical Value of the quantity.

There must be as many different units as there are different kinds of quantities to be measured, but in all dynamical sciences it is possible to define these units in terms of the three funda- mental units of Length, Time, and Mass. Thus the units of area and of volume are defined respectively as the square and the cube whose sides are the unit of length.

Sometimes, however, we find several units of the same kind founded on independent considerations. Thus the gallon, or the volume of ten pounds of water, is used as a unit of capacity as well as the cubic foot. The gallon may be a convenient measure in some cases, but it is not a systematic one, since its numerical re- lation tcrthe cubic foot is not a round integral number.

2.] In framing a mathematical system we suppose the funda- mental units of length, time, and mass to be given, and deduce all the derivative units from these by the simplest attainable de- finitions.

The formulae at which we arrive must be such that a person VOL. i. B

2 PRELIMINARY. [3.

I

of any nation, by substituting for the different symbols the nu- merical values of the quantities as measured by his own national units, would arrive at a true result.

Hence, in all scientific studies it is of the greatest importance to employ units belonging to a properly defined system, and to know the relations of these units to the fundamental units, so that we may be able at once to transform our results from one system to another.

This is most conveniently done by ascertaining the dimensions of every unit in terms of the three fundamental units. When a given unit varies as the nth power of one of these units, it is said to be of n dimensions as regards that unit.

For instance, the scientific unit of volume is always the cube whose side is the unit of length. If the unit of length varies, the unit of volume will vary as its third power, and the unit of volume is said to be of three dimensions with respect to the unit of length.

A knowledge of the dimensions of units furnishes a test which ought to be applied to the equations resulting from any lengthened investigation. The dimensions of every term of such an equa- tion, with respect to each of the three fundamental units, must be the same. If not, the equation is absurd, and contains some error, as its interpretation would be different according to the arbi- trary system of units which we adopt *.

The Three Fundamental Units.

3.] (1) Length. The standard of length for scientific purposes in this country is one foot, which is the third part of the standard yard preserved in the Exchequer Chambers.

In France, and other countries which have adopted the metric system, it is the metre. The metre is theoretically the ten mil- lionth part of the length of a meridian of the earth measured from the pole to the equator ; but practically it is the length of a standard preserved in Paris, which was constructed by Borda to correspond, when at the temperature of melting ice, with the value of the preceding length as measured by Delambre. The metre has not been altered to correspond with new and more accurate measurements of the earth, but the arc of the meridian is estimated in terms of the original metre.

• The theory of dimensions was first stated by Fourier, TUorie de Ckaleur, § 160.

5-] THE THREE FUNDAMENTAL UNITS. 3

In astronomy the mean distance of the earth from the sun is sometimes taken as a unit of length.

In the present state of science the most universal standard of length which we could assume would be the wave length in vacuum of a particular kind of light, emitted by some widely diffused sub- stance such as sodium, which has well-defined lines in its spectrum. Such a standard would be independent of any changes in the di- mensions of the earth, and should be

their writings to be more permanent than that body.

In treating of the dimensions of units we shall call the unit of length [Z]. If I is the numerical value of a length, it is under- stood to be expressed in terms of the concrete unit [Z], so that the actual length would be fully expressed by I [Z].

4.] (2) Time. The standard unit of time in all civilized coun- tries is deduced from the time of rotation of the earth about its axis. The sidereal day, or the true period of rotation of the earth, can be ascertained with great exactness by the ordinary observa- tions of astronomers ; and the mean solar day can be deduced from this by our knowledge of the length of the year.

The unit of time adopted in all physical researches is one second of mean solar time.

In astronomy a year is sometimes used as a unit of time. A more universal unit of time might be found by taking the periodic time of vibration of the particular kind of light whose wave length is the unit of length.

We shall call the concrete unit of time [T7], and the numerical measure of time I.

5.] (3) Mass. The standard unit of mass is in this country the avoirdupois pound preserved in the Exchequer Chambers. The grain, which is often used as a unit, is defined to be the 7000th part of this pound.

In the metrical system it is the gramme, which is theoretically the mass of a cubic centimetre of distilled water at standard tem- perature and pressure, but practically it is the thousandth part of the standard kilogramme preserved in Paris.

The accuracy with which the masses of bodies can be com- pared by weighing is far greater than that hitherto attained in the measurement of lengths, so that all masses ought, if possible, to be compared directly with the standard, and not deduced from experiments on water.

In descriptive astronomy the mass of the sun or that of the

B 2

4 PRELIMINAKY. [5.

earth is sometimes taken as a unit, but in the dynamical theory of astronomy the unit of mass is deduced from the units of time and length, combined with the fact of universal gravitation. The astronomical unit of mass is that mass which attracts another body placed at the unit of distance so as to produce in that body the unit of acceleration.

In framing a universal system of units we may either deduce the unit of mass in this way from those of length and time already denned, and this we can do to a rough approximation in the present state of science ; or, if we expect * soon to be able to determine the mass of a single molecule of a standard substance, we may wait for this determination before fixing a universal standard of mass.

We shall denote the concrete unit of mass by the symbol _M ] in treating of the dimensions of other units. The unit of mass will be taken as one of the three fundamental units. When, as in the French system, a particular substance, water, is taken as a standard of density, then the unit of mass is no longer inde- pendent, but varies as the unit of volume, or as [^3].

If, as in the astronomical system, the unit of mass is defined with respect to its attractive power, the dimensions of [M ] are [&T-2].

For the acceleration due to the attraction of a mass m at a

distance r is by the Newtonian Law — . Suppose this attraction

to act for a very small time t on a body originally at rest, and to cause it to describe a space s, then by the formula of Galileo,

rs whence m — 2 — g- . Since r and s are both lengths, and t is a

time, this equation cannot be true unless the dimensions of 9* are [Jy3!7"2]. The same can be shewn from any astronomical equa- tion in which the mass of a body appears in some but not in all of the terms f.

• See Prof. J. Loschmidt, ' Zur Grosse der Luftmolecule,' Academy of Vienna, Oct. 12, 1865 ; G. J. Stoney on ' The Internal Motions of Gases,' Phil. Mag., Aug. 1868 ; and Sir W. Thomson on ' The Size of Atoms,' Nature, March 31, 1870.

t If a centimetre and a second are taken as units, the astronomical unit of mass

would be about 1'537 x 107 grammes, or 15'37 tonnes according to Baily's repetition

,.-/r of Cavendish's experiment. Baily adopts 5'6604 as the result of all his experiments

• as the ^mean density of the earth, and this, with the values used by Baily for the

dimensions of the earth and the intensity of gravitation at its surface, gives the above

r gj^ r >*alue as the direct result of his experiments.

•  !'--^«,^«.«^Jw         /        &U*.e*       *?i*r&*,      &**S&£.          >-/';»    C*<*f&tA^ 
  

6.] DERIVED UNITS. 5

Derived Units.

6.] The unit of Velocity is that velocity in which unit of length is described in unit of time. Its dimensions are [Z77"1].

If we adopt the units of length and time derived from the vibrations of light, then the unit of velocity is the velocity of light.

The unit of Acceleration is that acceleration in which the velo- city increases by unity in unit of time. Its dimensions are _I/T~2].

The unit of Density is the density of a substance which contains unit of mass in unit of volume. Its dimensions are [3/Jy~3].

The unit of Momentum is the momentum of unit of mass moving with unit of velocity. Its dimensions are \MLTl.

The unit of Force is the force which produces unit of momentum in unit of time. Its dimensions are [MLT2].

This is the absolute unit of force, and this definition of it is implied in every equation in Dynamics. Nevertheless, in many text books in which these equations are given, a different unit of force is adopted, namely, the weight of the national unit of mass; and then, in order to satisfy the equations, the national unit of mass is itself abandoned, and an artificial unit is adopted as the dynamical unit, equal to the national unit divided by the numerical value of the intensity of gravity at the place. In this way both the unit of force and the unit of mass are made to depend on the value of the intensity of gravity, which varies from place to place, so that state- ments involving these quantities are not complete without a know- ledge of the intensity of gravity in the places where these statements were found to be true.

The abolition, for all scientific purposes, of this method of measur- ing forces is mainly due to the introduction by Gauss of a general system of making observations of magnetic force in countries in which the intensity of gravity is different. All such forces are now measured according to the strictly dynamical method deduced from our definitions, and the numerical results are the same in whatever country the experiments are made.

The unit of Work is the work done by the unit of force acting through the unit of length measured in its own direction. Its dimensions are [ML2T2.

The Energy of a system, being its capacity of performing work, is measured by the work which the system is capable of performing by the expenditure of its whole energy.

6 PRELIMINARY. [7.

The definitions of other quantities, and of the units to which they are referred, will be given when we require them.

In transforming the values of physical quantities determined in terms of one unit, so as to express them in terms of any other unit of the same kind, we have only to remember that every expres- sion for the quantity consists of two factors, the unit and the nu- merical part which expresses how often the unit is to be taken. Hence the numerical part of the expression varies inversely as the magnitude of the unit, that is, inversely as the various powers of the fundamental units which are indicated by the dimensions of the derived unit.

On Physical Continuity and Discontinuity.

7.] A quantity is said to vary continuously if, when it passes from one value to another, it assumes all the intermediate values.

We may obtain the conception of continuity from a consideration of the continuous existence of a particle of matter in time and space. Such a particle cannot pass from one position to another without describing a continuous line in space, and the coordinates of its position must be continuous functions of the time.

In the so-called ' equation of continuity,' as given in treatises on Hydrodynamics, the fact expressed is that matter cannot appear in or disappear from an element of volume without passing in or out through the sides of that element.

A quantity is said to be a continuous function of its variables if, when the variables alter continuously, the quantity itself alters continuously.

Thus, if u is a function of #, and if, while x passes continuously from XQ to #15 u passes continuously from UQ to ul9 but when x passes from ^ to #2, u passes from u{ to u2, u± being different from «!, then u is said to have a discontinuity in its variation with respect to x for the value #=#1} because it passes abruptly from u± to %' while x passes continuously through x±.

If we consider the differential coefficient of u with respect to x for the value 3?=^ as the limit of the fraction

when x.2 and #0 are both made to approach xl without limit, then, if XQ and x2 are always on opposite sides of asl9 the ultimate value of the numerator will be <— wlf and that of the denominator will be zero. If u is a quantity physically continuous, the discontinuity

8.] CONTINUITY AND DISCONTINUITY. 7

can exist only with respect to the particular variable x. "We must in this case admit that it has an infinite differential coefficient when X=XL. If u is not physically continuous, it cannot be dif- ferentiated at all.

It is possible in physical questions to get rid of the idea of discontinuity without sensibly altering the conditions of the case. If XQ is a very little less than jcl9 and #2 a very little greater than a?1} then u0 will be very nearly equal to % and u2 to u{. We may now suppose u to vary in any arbitrary but continuous manner from UQ to ?/2 between the limits #0 and #2. In many physical questions we may begin with a hypothesis of this kind, and then investigate the result when the values of XQ and #2 are made to approach that of x± and ultimately to reach it. If the result is independent of the arbitrary manner in which we have supposed u to vary between the limits, we may assume that it is true when u is discontinuous.

Discontinuity of a Function of more than One Variable.

8.] If we suppose the values of all the variables except x to be constant, the discontinuity of the function will occur for particular values of #, and these will be connected with the values of the other variables by an equation which we may write

4> = $ (»,#*,&».) = 0.

The discontinuity will occur when $ — 0. When $ is positive the function will have the form F2 (x,y> z, &c.). When <£ is negative it will have the form F1 (x,y, #, &c.). There need be no necessary relation between the forms Fl and F2.

To express this discontinuity in a mathematical form, let one of the variables, say #, be expressed as a function of $ and the other variables, and let F1 and F2 be expressed as functions of $,y, z, &c. We may now express the general form of the function by any formula which is sensibly equal to F2 when <£ is positive, and to F1 when (/> is negative. Such a formula is the following —

As long as n is a finite quantity, however great, F will be a continuous function, but if we make n infinite F will be equal to F2 when <£ is positive, and equal to Fl when <£ is negative.

8 PRELIMINARY. [9.

Discontinuity of the Derivatives of a Continuous Function.

The first derivatives of a continuous function may be discon- tinuous. Let the values of the variables for which the discon- tinuity of the derivatives occurs be connected by the equation

<J> = <f>(0,#*...) = 0,

and let F1 and F2 be expressed in terms of <£ and n — 1 other variables, say (y, z . . .).

Then, when <£ is negative, Fl is to be taken, and when $ is positive F2 is to be taken, and, since F is itself continuous, when d> is zero, F, = F2*

J TJ J Jji

Hence, when d> is zero, the derivatives — r-^ and —~ may be

different, but the derivatives with respect to any of the other

7 TJ J ~fjl

variables, such as —^ and — - , must be the same. The discon-

dy dy

«/ j

tinuity is therefore confined to the derivative with respect to $, all the other derivatives being continuous.

Periodic and Multiple Functions.

9.] If u is a function of x such that its value is the same for x, x + a, x --nctj and all values of x differing by a, u is called a periodic function of #, and a is called its period.

If x is considered as a function of u, then, for a given value of M, there must be an infinite series of values of x differing by. multiples of a. In this case x is called a multiple function of u, and a is called its cyclic constant.

fj rp

The differential coefficient — has only a finite number of values

du

corresponding to a given value of u.

On the Relation of Physical Quantities to Directions in Space. 10.] In distinguishing the kinds of physical quantities, it is of great importance to know how they are related to the directions of those coordinate axes which we usually employ in defining the positions of things. The introduction of coordinate axes into geo- metry by Des Cartes was one of the greatest steps in mathematical progress, for it reduced the methods of geometry to calculations performed on numerical quantities. The position of a point is made to depend on the length of three lines which are always drawn in determinate directions, and the line joining two points is in like manner considered as the resultant of three lines.

II.] VECTORS, OR DIRECTED QUANTITIES. 9

But for many purposes of physical reasoning1, as distinguished from calculation, it is desirable to avoid explicitly introducing- the Cartesian coordinates, and to fix the mind at once on a point of space instead of its three coordinates, and on the magnitude and direction of a force instead of its three components. This mode of contemplating- geometrical and physical quantities is more prim- itive and more natural than the other, although the ideas connected with it did not receive their full development till Hamilton made the next great step in dealing with space, by the invention of his Calculus of Quaternions.

As the methods of Des Cartes are still the most familiar to students of science, and as they are really the most useful for purposes of calculation, we shall express all our results in the Cartesian form. I am convinced, however, that the introduction of the ideas, as distinguished from the operations and methods of Quaternions, will be of great use to us in the study of all parts of our subject, and especially in electrodynamics, where we have to deal with a number of physical quantities, the relations of which to each other can be expressed far more simply by a few expressions of Hamilton's, than by the ordinary equations.

11.] One of the most important features of Hamilton's method is the division of quantities into Scalars and Vectors.

A Scalar quantity is capable of being completely defined by a single numerical specification. Its numerical value does not in any way depend on the directions we assume for the coordinate axes.

A Vector, or Directed quantity, requires for its definition three numerical specifications, and these may most simply be understood as having reference to the directions of the coordinate axes.

Scalar quantities do not involve direction. The volume of a geometrical figure, the mass and the energy of a material body, the hydrostatical pressure at a point in a fluid, and the potential at a point in space, are examples of scalar quantities.

A vector quantity has direction as well as magnitude, and is such that a reversal of its direction reverses its sign. The dis- placement of a point, represented by a straight line drawn from its original to its final position, may be taken as the typical vector quantity, from which indeed the name of Vector is derived.

The velocity of a body, its momentum, the force acting on it, an electric current, the magnetization of a particle of iron, are instances of vector quantities.

10 PKELIMINARY. [l2.

There are. physical quantities of another kind which are related to directions in space, but which are not vectors. Stresses and strains in solid bodies are examples of these, and so are some of the properties of bodies considered in the theory of elasticity and in the theory of double refraction. Quantities of this class require for their definition nine numerical specifications. They are ex- pressed in the language of Quaternions by linear and vector functions of a vector.

The addition of one vector quantity to another of the same kind is performed according to the rule given in Statics for the com- position of forces. In fact, the proof which Poisson gives of the 'parallelogram of forces' is applicable to the composition of any quantities such that turning them end for end is equivalent to a reversal of their sign.

When we wish to denote a vector quantity by a single symbol, and to call attention to the fact that it is a vector, so that we must consider its direction as well as its magnitude, we shall denote it by a German capital letter, as $1, S3, &c.

In the calculus of Quaternions, the position of a point in space is defined by the vector drawn from a fixed point, called the origin, to that point. If we have to consider any physical quantity whose value depends on the position of the point, that quantity is treated as a function of the vector drawn from the origin. The function may be itself either scalar or vector. The density of a body, its temperature, its hydrostatic pressure, the potential at a point, are examples of scalar functions. The resultant force at a point, the velocity of a fluid at a point, the velocity of rotation of an element of the fluid, and the couple producing rotation, are examples of vector functions.

12.] Physical vector quantities may be divided into two classes, in one of which the quantity is defined with reference to a line, while in the other the quantity is defined with reference to an area.

For instance, the resultant of an attractive force in any direction may be measured by finding the work which it would do on a body if the body were moved a short distance in that direction and dividing it by that short distance. Here the attractive force is defined with reference to a line.

On the other hand, the flux of heat in any direction at any point of a solid body may be defined as the quantity of heat which crosses a small area drawn perpendicular to that direction divided

13.] INTENSITIES AND FLUXES. 11

by that area and by the time. Here the flux is defined with reference to an area.

There are certain cases in which a quantity may be measured with reference to a line as well as with reference to an area.

Thus, in treating of the displacements of elastic solids, we may direct our attention either to the original and the actual position of a particle, in which case the displacement of the particle is measured by the line drawn from the first position to the second, or we may consider a small area fixed in space, and determine what quantity of the solid passes across that area during the dis- placement.

In the same way the velocity of a fluid may be investigated either with respect to the actual velocity of the individual particles, or with respect to the quantity of the fluid which flows through any fixed area.