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Book DetailsAMS Chelsea PublishingVolume: 237; 1971; 547 ppMSC: Primary 78; 01
Oliver Heaviside is probably best known to the majority of mathematicians for the Heaviside function in the theory of distribution. However, his main research activity concerned the theory of electricity and magnetism, the area in which he worked for most of his life. Results of this work are presented in his fundamental threevolume Electromagnetic Theory. The book brings together many of Heaviside's published and unpublished notes and short articles written between 1891 and 1912. One of Heaviside's main achievements was the recasting of Maxwell's theory of electromagnetism into the form currently used by everyone. He is also known for the invention of operational calculus and for major contributions to solving theoretical and practical problems of cable and radio communication. All this is collected in three volumes of Electromagnetic Theory. However, there is even more. For example, Chapter V in Volume II discusses the age of Earth, and several sections in Volume III talk about the teaching of mathematics in school.
In addition to Heaviside's writings, two detailed surveys of Heaviside's work, by Sir Edmund Whittaker and by B. A. Behrend, are included in Volume I, and a long account of Heaviside's unpublished notes (which he presumably planned to publish as Volume IV of Electromagnetic Theory) is included in Volume III.
This item is also available as part of a set: 
Table of Contents

Front Cover

PREFACE TO VOL. II.

CONTENTS OF VOLUME II.

CHAPTER V. MATHEMATICS AND THE AGE OF THE EARTH.

§ 223. Mathematics is an Experimental Science.

§ 224. Rigorous Mathematics is Narrow, Physical Mathematics Bold and Broad.

§ 225. Physical Problems lead to Improved Mathematical Methods.

§ 226. "Mathematicsand Mathematics." Remarkable Phenomenon.

§ 227. The Age of the Earth. Kelvin's Problem.

§ 228. Perry's Modification. Remarkable Result.

§ 229. Cooling of a.n Infinite Block composed of Two Materials.

§ 230. Large Correction for Sphericity in Perry's Problem.

§ 231. Remarks on the Age of the Earth.

§ 232. Peculiar Nature of the Problem of the Cooling of a Homogeneous Sphere with a Resisting Skin.

§ 233. Cooling of a. Body of Variable Conductivity and Capacity but with their Product Constant.

§ 234. Magnitude of the Correction for Sphericity in Various Cases.

§ 235. Explanation of the last.

§ 236. Investigation by the Wave Method of the Cooling of a Homogeneous Sphere with a Resisting Skin. Effect of Varying the Constants.

§ 237. Importance of the Operational Method.

CHAPTER VI. PURE DIFFUSION OF ELECTRIC DISPLACEMENT.

§ 238. Analogy between the Diffusion of Heat in a Rod and the Diffusion of Charge in a Cable.

§ 239. The Operational Method assists Fourier.

§ 240. The Characteristic Equation and Solution in terms of TimeFunctions.

§ 241. Steady Impressed Force at Beginning of Cable. Fractional Differentiation. Simply Periodic Force.

§ 242. Effect of a Terminal Arrangement. Two Solutions in the Case of a Resistance.

§ 243. Theory of a Terminal Condenser.

§ 244. Theory of a Terminal Inductance.

§ 245. The General Nature of Electrical Operators.

§ 246. The Simple Waves of Potential and Current.

§247. The Error Function. Short Table.

§ 248. The Way the Charge and Current Spread.

§ 249. Theory of an Impulsive Current produced by a Continued Impressed Force.

§ 250. Diffusion of a Charge initially at One Point. Arbitrary Source of Electrification.

§ 251. The Inversion of Operators. Simple Examples.

§ 252. The E:ffect of a. Steady Current impressed at the Origin.

§ 253. Nature and Effect of Multiple Impulses.

§ 254. Convenient Way of denoting Diffusion Formulæ

§ 255. Reflected Waves. Cases of Simple Reflection.

§ 256. An Infinite Series of Reflected Waves. Line Earthed at Both Ends.

§ 257. The Method of Images. The Waves are really Successive.

§ 258. Reflection at an Insulated Terminal.

§ 259. General Case. Effect of an Impressed Force at an Intermediate Point, with any Terminal Conditions.

§ 260. Four Cases of Elementary Waves.

§ 261. The Reflection Coefficients in terms of the Terminal Resistance Operators.

§ 262. Cases of Vanishing or Constancy of the ReflectionCoefficients.

§ 263. GeneraI Case of an Intermediate Source of Electrification subject to any Terminal Conditions.

§ 264. The Two Ways of expressing Propagational Results, in terms of Waves, or of Vibrations.

§ 265. Conversion of Operational Solutions to Fourier Series by Special Ways. (1). Effect due to e at A, when earthed at A and B.

§ 266. (2). Modified Way of doing the Last Case.

§ 267. (3). Earth at Both Terminals. Initial Charge at a Point. Arbitrary Initial State.

§ 268. (4). Line Cut at Both Terminals. Effect of Impressed Force.

§ 269. (5). Line Cut at Terminals. Effect of Initial Charge.

§ 270. (6). Earth at A and Cut at B. Effect of Initial Charge.

§ 271. Periodic Expression of Impulsive Functions. Fourier's Theorem.

§ 272. Fourier Series in General. Various Sorts Needed even when the Function is Cyclic and Continuous. Expansions of Zero.

§ 273. Special Forms of Fourier Integrals. Interchangeable Property. Use in Transforming Definite Integrals.

§ 274. Continuous Passage from Wave Series to Fourier Series, and its Reversion.

§ 275. How to :find the Meaning of a Fourier Series Operationally.

§ 276. Taylor's Theorem in its Essentials Operationally Considered.

§ 277. A Fourier Series involving a Parabola interpreted by Taylor's Theorem.

§ 278. Representation of a Row of Impulses by Taylor's Theorem, leading to Fourier's.

§ 279. Laurent's Theorem and Fourier's.

§ 280. On Operational Solutions and their Interpretation.

§ 281. Sketch of Way of extending Fourier's Method to Fourier Series in General.

§ 282. The Expansion Theorem. Operational Way of getting Exnansions in Normal Functions.

§ 283. Examples of the Use of the Expansion Theorem: (1). Inductance Coil and Condenser separately.

§ 284. The Treatment of Simply Periodic Cases.

§ 285. (2). Coil and Condenser in Sequence. Also in Parallel

§ 286. (3). Two Coils under Mutual Influence.

§ 287. (4). Cable Earthed at A and B. Impressed Force at A.

§ 288. (5). Cable Earthed at A and Cut at B. Impressed Force at A.

§ 289. (6). Earth at A, Cut at B. Impressed Force at y.

§ 290. (7). Earth at A and B. Impressed Force at y.

§ 291. (8). General Terminal Conditions. Impressed Force a.t A.

§ 292. (9). General Case of an Intermediate Impressed Force.

§ 293. The Determinantal Equation.

§293a. Subsidence of Special Initial States.

§ 294. (10). General Case of an Arbitrary Initial State in the Cable.

§ 295. (11). Auxiliary Expansions due to the Terminal Energy. Case of a Condenser

§ 296. Points of Infinite Condensation. Exceptions to Fourier's Theorem.

§ 297. Abnormal Fourier Series.

§ 298. Origin of Two Principal Abnormal Cases.

§ 299. First General Case : Z0 =  Z1•

§ 300. Equal Positive and Negative Terminal Resistances.

§ 301. Equal Positive and Negative Terminal Permittances.

§ 302. Physical Interpretation of the Abnormal Case of § 300.

§ 303. Positive Terminal Resistance and Negative Terminal Permittance.

§ 304. Equal Positive and Negative Terminal Inductances.

§ 305. Impressed Force in the Case Z0 =  Z1.

§ 306. Singular Extreme Case of Z0 or  Z1 being a Cable equivalent to the Main One.

§ 307. More Genera.I Case to Elucidate the Last. Terminal Cable Z0 not equivalent to Main One.

§ 308. Arbitrary Initial State when Z0 or  Z1 is a Cable. Singular Case.

§ 309. Real Terminal Conditions. Terminal Arbitraries Case of a Coil. Two Ways of Treatment.

§ 310. Terminal Coil and Condenser in Sequence.

§ 311. Coil and Condenser in Parallel.

§ 312. Two Coils in Sequence or in Parallel.

§ 313. A Closed Cable with a Leak. Split into Two Simpler Cases.

§ 314. Closed Cable and Leak. Another Way.

§ 315. Closed Cable with Intermediate Insertion. Split into Two Simpler Cases.

§ 316. Same as last without Initial Splitting.

§ 317. Closed Cable with Discontinuous Potential and Current.

§ 318. A Cable in Closed Circuit without Constraint.

§ 319.Theory of a Leak. Normal Systems.

§ 320. Theory of a Leak. Operational Solution.

§ 321. Evaluation of Energy in Normal Systems.

§ 322. Initial States in Combinations.

§ 323. Two Cables with different Constants in sequence, with an insertion.

§ 324. Cable involving the zeroth Bessel function.

§ 325. A Fourier and a Bessel Cable in sequence.

§ 326. Construction of a Normal System in General.

§ 327. Construction of Operational Solutions in a. Connected System.

§ 328. Remarks on Operational and Normal Solutions. Connection with the Simply Periodic.

§ 329. A Bessel Cable with one Terminal condition. Two Bessel Cables in Sequence.

§ 330. General Solutions for Sources in a Bessel Cable with two Terminal conditions.

§ 332. Operational Solutions for Sources in the General Case.

§ 333. Conversion of Wire Waves to Cylindrical Waves. Two Ways.

§ 334. Special Cases of Zeroth Bessel Solutions.

§ 335. Numerical Interpretation of Formulæ. The Divergent Series.

§ 336. The divergent Formulæ are fundamental. Generation of Waves in a Medium whose Constants vary as the nth power of the distance.

§ 337. Construction of General Solution by Waves.

§ 338. Construction of General Solution by the Convergent Formulæ.

§ 339. Comparison of Wave and Vibrational Solutions to deduce Relation of Divergent to Convergent Bessel functions.

§ 340. Nature of Algebraical Transformation from Divergent to Convergent Formulæ.

§ 341. Rationality in p of Operational Solutions with two Boundaries. Solutions in terms of Im and Km.

§ 342. The convergent Oscillating Bessel functions, and Operational Solutions in terms thereof.

§ 343. The divergent Oscillating Bessel functions.

§ 344. Physical reason of the unlikeness of the two divergent functions Hm, Km.

§ 345. Electrical Argument showing the Impotency of Restraints at the origin, unless 1 >n> 1.

§ 346. Reduced Formulæ when one Boundary is at the Origin.

§ 347. The Expansion Theorem and Bessel Series. The Potential due to initial Charge.

§ 348. Time Function when Selfinduction is allowed for.

§ 349. The Potential due to initial Current.

§ 349A. Uniform Subsidence of Induction a.nd Displacement in combinations of Coils and Condensers.

§ 349 B. Uniform Subsidence of Mean Voltage in a. Bessel Circuit.

§ 349c. Uniform Subsidence of Mean Current in a Bessel Circuit.

APPENDIX C. RATIONAL UNITS.

CHAPTER VII. ELECTROMAGNETIC WAVES AND GENERALISED DIFFERENTIATION.

§ 350. Determination of the Value of pH by a Diffusion Problem.

§ 351. Elementary generalised differentiation. Value of pml when m is integral or midway between.

§ 352. Cable Problem:C=(K+Sp)½(R1+Lp)½V (1 ). Elementary Cases by Inspection.

§ 353. (2). Algebrisation when e is constant and K zero. Two ways. Convergent and divergent results.

§ 354. (3). Third way. Change of Operand.

§ 355. (4). V due to steady C whenR=O. Instantaneous Impedance and Admittance.

§ 356. (5). C due to steady V when S = O. The error function again.

§ 357. (6). V due to steady C when L=O. Two ways.

§ 358. (7). V due to steady C when S is zero. Two ways.

§ 359. (8). All constants finite. C due to V varying as Ept_

§ 360. (9). C due to V varying as ∈Kt/S, Discharge of a charged into an empty Cable.

§ 361. (10). C due to steady V, and V due to steady C.

§ 362. {11). C due to impulsive V, and V due to impulsive C.

§ 363. Cubic under radical. Reversibility of Operations. Distribution of Operators.

§ 364. (12). Development of Equation (60). C due to steady V.

§ 365. (13). Another Development of Equation (60).

§ 366. (14). Third Development of Equation (60). Integration by Parts.

§ 367. (15). Generalisation. The Complete Wave of C due to V at the origin varying as ∈Kt/S,

§ 368. (16). Summary of Work showing the Wave of C due to V at origin varying as ∈ Kt/s.

§ 369.(17). Derivation of the Wave of V from the Wave of C.

§ 370. (18). The Wave of V independently developed.

§ 371. (19). The Waves of V and C due to Initial Charge or Momentum at a Single Place.

§ 372. The Waves of V and C due to a Steady Voltage Impressed at the Origin.

§ 373. Analysis of Transmission Operator to Show the Deformation, Progression and Attenuation of Waves.

§ 374. Three Examples. The Wave of V due to impressed Voltage, varying as ∈  pt, ∈ Kt/S, or steady.

§ 375. Expansion of Distortion Operators in Powers of p1.

§ 376. Example. The Current Wave due to Impressed Voltage e0∈Kt/S at the Origin.

§ 377. Value of (p2  σ 2)*lo(σt) when n is a Positive or Negative Integer. Structure of the Convergent Bessel Functions.

§ 378. Analysis Founded upon the Division of the Instantaneous State into Positive and Negative Pure Waves.

§ 379. Simplest Solutions. Waves of Infinite Length, and of Length 2πv/σ.

§ 380. Development of GeneraI Solution in ump m and wmP m Functions.

§ 381. Derivation of C Wave from V Wave, and Conversely, with Examples. Condition at a Moving Boundary. Expansion Of ∈σ t .

§ 382. Deduction of the V and C Waves when VO is Constant from the Case VO = e∈  Kt/S. Expansion of ∈Pt in I Functions. Construction of the Wave of V due to any Impressed Voltage.

§ 383. Identical Expansions of Functions in In Functions. Formula for (½x)n. Electromagnetic Applications.

§ 384. Expansion of any Power Series in I,. Functions. Examples.

§ 385. Expansion of a Power Series in Jn Functions. Examples.

§ 386. The Waves of V and C due to an::v V0 developed in w"'Pm(z) Functions from the Operational Form of V 0∈pt.

§ 387. Remarkable Formula for the Expansion of a. Function in In Functions, and Examples. Modification of § 386, and Example.

§ 388 Impulsive Impressed Voltages, and the Impulsive Waves and their Tails generated.

§ 389. Tendency of Distortion to vanish in rapid Fluctuations. Effect of increased Resistance in rounding off corners and distorting.

§ 390. Wave Due to Impressed Voltage varying as ∈pt t1.

§ 391. Wave Due to V0 varying as ∈Ptlog t.

§ 392. Effect of a Terminal Resistance as expected in 1887 and as found in 1896.

§ 393. Reflection at the Free Ends of a Wire. A Series of Spherical Waves.

§ 394. Reflection at the End of a Circuit terminated by a Plane Resisting Sheet.

§ 395. Long Wave Formulie for Terminal Redaction.

§ 396. Reflection of Long Waves in General.

§ 397. Terminal Reflection without Loss. Wave Solutions.

§ 398. Comparison with Fourier Series. Solution of Definite Integrals.

§ 399. General Way of Finding Second and Following Waves due to Terminal Reflection.

§ 400. Application to Terminal Resistances. Full Solutions witb the Critical Resistances. Second Wave with any Resistance.

§ 401. Inversion of Operations. Derivation of First Wave from the Second.

§ 402. Derivation of Third and Later Waves from the Second.

§ 403. Summarised Complete Solutions.

§ 404. Reflection by a Condenser.

§ 405. Reflection by an Inductance Coil.

§ 406. Initial States. Expression of Results by Definite Integrals.

§ 407. The Special Initial States J0 (σx/v) and Jn (σx/v).

§ 408. The States of V and C resulting from any Initial States, V0 and C0, expansible in Jn functions.

§ 409. Some Fundamental Examples.

§ 410. The Genera.I Solution for any Initial State, and some Simple Examples.

§ 411. Conversion to Definite Integrals. Short Cut to Fourier's Theorem.

§ 412. The Space Integrals of V and C, due to Elements a.t the Origin.

§ 413. The Time Integral of C, due to Elements at the Origin.

§414. Evaluation of the Fundamental Integral.

§ 415. Generalisation of the Integral. Both kinds of Bessel Functions.

§ 416. The C due to initial V0 • Operational Method, and Modification.

§ 417. The V due to initial V0.

§ 418. Final Investigation of the V and C due to initial V0 and C0.

§ 419. Undistorted Waves without and with attenuation.

§ 420. Effects of Resistance and Leakance on an Initial State of Constant VO on one side of the Origin.

§ 421. Division of Charge Initially at the Origin into Two Waves with Positive or Negative Charge between them.

§ 422. The Current due to Initial Charge on one side of the Origin.

§ 423. The Aftereffects of an Initially Pure Wave. Positive and Negative Tails.

§ 424. Figs. 1 to 13 described in terms of Electromagnetic Waves in a doubly Conducting Medium.

CHAPTER VIII. GENERALISED DIFFERENTIATION AND DIVERGENT SERIES.

§ 425. A Formula for n obtained by Harmonic Analysis.

§426. Algebraical Construction of g(n). Value of g(n)g(  n).

§ 427. Generalisation of Exponential Function.

§ 428. Application of the Generalised Exponential to & Bessel Function, to the Binomial Theorem, and to Taylor's Theorem.

§ 429. Algebraical Connection of the Convergent and Divergent Series for the Zeroth Bessel Function.

§ 430. Limiting Form of Generalised Binomial Expansion when Index is 1.

§ 431. Remarks on the Operator (I+ Ll1)n.

§ 432. Remarks on the Use of Divergent Series.

§ 433. Logarithmic Formula derived from Binomial.

§ 434. Logarithmic Formulm derived from Generalised Exponential.

§ 435. Connections of the Zeroth Bessel Functions.

§ 436. Operational Properties of the Zeroth Bessel Functions.

§ 437. Remarks on Common and Generalized Mathematics.

§ 438. The Generalised Zeroth Bessel Function Analysed.

§ 439. Expression of the Divergent Ho{x) a.nd K0 (x) in Terms of Two Generalised Bessel Functions. Generalisation of ∈x.

§ 440. The Divergent Hn(x) and Kn(x) in Terms of Two Special Generalised Bessel Functions.

§ 441. The Divergent Hn(x) and Kn(x) in Terms of any Generalised Bessel Function of the same order.

§ 442. Product of the Series for ∈x and ∈x cos πr. Possible Transition from ∈x to ∈x.

§ 443. Power Series for log x.

§ 444. Examination of some Apparent Equivalences, and Rectification.

§ 445. Determination of the Meaning of a Generalised Bessel Function in terms of H0 and K0

§ 446. Some Apparent Equivalences.

§ 447. Cotangent Formula and Derived Formula for Logarithm. Various Properties of these and other Divergent Series.

§ 448. Three Electrical Examples of Equivalent Convergent and Divergent Series.

§ 449. Sketch of Theory of Algebrisation of (1 bp r)1 1.

APPENDICES.

APPENDIX D. ON COMPRESSIONAL ELECTRIC OR MAGNETIC WAVES.

APPENDIX E. DISPERSION.

APPENDIX F. ON THE TRANSFORMATION OF OPTICAL WAVE SURFACES BY HOMOGENEOUS STRAIN.*

1. Simplex Eolotropy.

2. Properties connected with Duplex Eolotropy.

3. Effects of straining a Duplex Wave Surface.

4. Forms of the Index and WaveSurface Equations, and theProperties of Inversion and Interchangeability of Operators.

5. General Transformation of WaveSurface by Homogeneous Strain.

6. Special Cases of Reduction to a Simplex WaveSurface.

7. Transformation from Duplex Wave to IndexSurface by aPure Strain.

8. Substitution of three successive Pure Straws for one .Two ways .

9. Transformation of Characteristic Equation by Strain.

10. Derivation of Index Equation from Characteristic.

APPENDIX G. NOTE ON THE MOTION OF A CHARGED BODY AT A SPEED EQUAL TO OR GREATER THAN THAT OF LIGHT.

APPENDIX H. NOTE ON ELECTRICAL WAVES IN SEA WATER.*

APPENDIX I. NOTE ON THE ATTENUATION OF HERTZIAN WAVES ALONG WIRES.*

NOTES.

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Oliver Heaviside is probably best known to the majority of mathematicians for the Heaviside function in the theory of distribution. However, his main research activity concerned the theory of electricity and magnetism, the area in which he worked for most of his life. Results of this work are presented in his fundamental threevolume Electromagnetic Theory. The book brings together many of Heaviside's published and unpublished notes and short articles written between 1891 and 1912. One of Heaviside's main achievements was the recasting of Maxwell's theory of electromagnetism into the form currently used by everyone. He is also known for the invention of operational calculus and for major contributions to solving theoretical and practical problems of cable and radio communication. All this is collected in three volumes of Electromagnetic Theory. However, there is even more. For example, Chapter V in Volume II discusses the age of Earth, and several sections in Volume III talk about the teaching of mathematics in school.
In addition to Heaviside's writings, two detailed surveys of Heaviside's work, by Sir Edmund Whittaker and by B. A. Behrend, are included in Volume I, and a long account of Heaviside's unpublished notes (which he presumably planned to publish as Volume IV of Electromagnetic Theory) is included in Volume III.

Front Cover

PREFACE TO VOL. II.

CONTENTS OF VOLUME II.

CHAPTER V. MATHEMATICS AND THE AGE OF THE EARTH.

§ 223. Mathematics is an Experimental Science.

§ 224. Rigorous Mathematics is Narrow, Physical Mathematics Bold and Broad.

§ 225. Physical Problems lead to Improved Mathematical Methods.

§ 226. "Mathematicsand Mathematics." Remarkable Phenomenon.

§ 227. The Age of the Earth. Kelvin's Problem.

§ 228. Perry's Modification. Remarkable Result.

§ 229. Cooling of a.n Infinite Block composed of Two Materials.

§ 230. Large Correction for Sphericity in Perry's Problem.

§ 231. Remarks on the Age of the Earth.

§ 232. Peculiar Nature of the Problem of the Cooling of a Homogeneous Sphere with a Resisting Skin.

§ 233. Cooling of a. Body of Variable Conductivity and Capacity but with their Product Constant.

§ 234. Magnitude of the Correction for Sphericity in Various Cases.

§ 235. Explanation of the last.

§ 236. Investigation by the Wave Method of the Cooling of a Homogeneous Sphere with a Resisting Skin. Effect of Varying the Constants.

§ 237. Importance of the Operational Method.

CHAPTER VI. PURE DIFFUSION OF ELECTRIC DISPLACEMENT.

§ 238. Analogy between the Diffusion of Heat in a Rod and the Diffusion of Charge in a Cable.

§ 239. The Operational Method assists Fourier.

§ 240. The Characteristic Equation and Solution in terms of TimeFunctions.

§ 241. Steady Impressed Force at Beginning of Cable. Fractional Differentiation. Simply Periodic Force.

§ 242. Effect of a Terminal Arrangement. Two Solutions in the Case of a Resistance.

§ 243. Theory of a Terminal Condenser.

§ 244. Theory of a Terminal Inductance.

§ 245. The General Nature of Electrical Operators.

§ 246. The Simple Waves of Potential and Current.

§247. The Error Function. Short Table.

§ 248. The Way the Charge and Current Spread.

§ 249. Theory of an Impulsive Current produced by a Continued Impressed Force.

§ 250. Diffusion of a Charge initially at One Point. Arbitrary Source of Electrification.

§ 251. The Inversion of Operators. Simple Examples.

§ 252. The E:ffect of a. Steady Current impressed at the Origin.

§ 253. Nature and Effect of Multiple Impulses.

§ 254. Convenient Way of denoting Diffusion Formulæ

§ 255. Reflected Waves. Cases of Simple Reflection.

§ 256. An Infinite Series of Reflected Waves. Line Earthed at Both Ends.

§ 257. The Method of Images. The Waves are really Successive.

§ 258. Reflection at an Insulated Terminal.

§ 259. General Case. Effect of an Impressed Force at an Intermediate Point, with any Terminal Conditions.

§ 260. Four Cases of Elementary Waves.

§ 261. The Reflection Coefficients in terms of the Terminal Resistance Operators.

§ 262. Cases of Vanishing or Constancy of the ReflectionCoefficients.

§ 263. GeneraI Case of an Intermediate Source of Electrification subject to any Terminal Conditions.

§ 264. The Two Ways of expressing Propagational Results, in terms of Waves, or of Vibrations.

§ 265. Conversion of Operational Solutions to Fourier Series by Special Ways. (1). Effect due to e at A, when earthed at A and B.

§ 266. (2). Modified Way of doing the Last Case.

§ 267. (3). Earth at Both Terminals. Initial Charge at a Point. Arbitrary Initial State.

§ 268. (4). Line Cut at Both Terminals. Effect of Impressed Force.

§ 269. (5). Line Cut at Terminals. Effect of Initial Charge.

§ 270. (6). Earth at A and Cut at B. Effect of Initial Charge.

§ 271. Periodic Expression of Impulsive Functions. Fourier's Theorem.

§ 272. Fourier Series in General. Various Sorts Needed even when the Function is Cyclic and Continuous. Expansions of Zero.

§ 273. Special Forms of Fourier Integrals. Interchangeable Property. Use in Transforming Definite Integrals.

§ 274. Continuous Passage from Wave Series to Fourier Series, and its Reversion.

§ 275. How to :find the Meaning of a Fourier Series Operationally.

§ 276. Taylor's Theorem in its Essentials Operationally Considered.

§ 277. A Fourier Series involving a Parabola interpreted by Taylor's Theorem.

§ 278. Representation of a Row of Impulses by Taylor's Theorem, leading to Fourier's.

§ 279. Laurent's Theorem and Fourier's.

§ 280. On Operational Solutions and their Interpretation.

§ 281. Sketch of Way of extending Fourier's Method to Fourier Series in General.

§ 282. The Expansion Theorem. Operational Way of getting Exnansions in Normal Functions.

§ 283. Examples of the Use of the Expansion Theorem: (1). Inductance Coil and Condenser separately.

§ 284. The Treatment of Simply Periodic Cases.

§ 285. (2). Coil and Condenser in Sequence. Also in Parallel

§ 286. (3). Two Coils under Mutual Influence.

§ 287. (4). Cable Earthed at A and B. Impressed Force at A.

§ 288. (5). Cable Earthed at A and Cut at B. Impressed Force at A.

§ 289. (6). Earth at A, Cut at B. Impressed Force at y.

§ 290. (7). Earth at A and B. Impressed Force at y.

§ 291. (8). General Terminal Conditions. Impressed Force a.t A.

§ 292. (9). General Case of an Intermediate Impressed Force.

§ 293. The Determinantal Equation.

§293a. Subsidence of Special Initial States.

§ 294. (10). General Case of an Arbitrary Initial State in the Cable.

§ 295. (11). Auxiliary Expansions due to the Terminal Energy. Case of a Condenser

§ 296. Points of Infinite Condensation. Exceptions to Fourier's Theorem.

§ 297. Abnormal Fourier Series.

§ 298. Origin of Two Principal Abnormal Cases.

§ 299. First General Case : Z0 =  Z1•

§ 300. Equal Positive and Negative Terminal Resistances.

§ 301. Equal Positive and Negative Terminal Permittances.

§ 302. Physical Interpretation of the Abnormal Case of § 300.

§ 303. Positive Terminal Resistance and Negative Terminal Permittance.

§ 304. Equal Positive and Negative Terminal Inductances.

§ 305. Impressed Force in the Case Z0 =  Z1.

§ 306. Singular Extreme Case of Z0 or  Z1 being a Cable equivalent to the Main One.

§ 307. More Genera.I Case to Elucidate the Last. Terminal Cable Z0 not equivalent to Main One.

§ 308. Arbitrary Initial State when Z0 or  Z1 is a Cable. Singular Case.

§ 309. Real Terminal Conditions. Terminal Arbitraries Case of a Coil. Two Ways of Treatment.

§ 310. Terminal Coil and Condenser in Sequence.

§ 311. Coil and Condenser in Parallel.

§ 312. Two Coils in Sequence or in Parallel.

§ 313. A Closed Cable with a Leak. Split into Two Simpler Cases.

§ 314. Closed Cable and Leak. Another Way.

§ 315. Closed Cable with Intermediate Insertion. Split into Two Simpler Cases.

§ 316. Same as last without Initial Splitting.

§ 317. Closed Cable with Discontinuous Potential and Current.

§ 318. A Cable in Closed Circuit without Constraint.

§ 319.Theory of a Leak. Normal Systems.

§ 320. Theory of a Leak. Operational Solution.

§ 321. Evaluation of Energy in Normal Systems.

§ 322. Initial States in Combinations.

§ 323. Two Cables with different Constants in sequence, with an insertion.

§ 324. Cable involving the zeroth Bessel function.

§ 325. A Fourier and a Bessel Cable in sequence.

§ 326. Construction of a Normal System in General.

§ 327. Construction of Operational Solutions in a. Connected System.

§ 328. Remarks on Operational and Normal Solutions. Connection with the Simply Periodic.

§ 329. A Bessel Cable with one Terminal condition. Two Bessel Cables in Sequence.

§ 330. General Solutions for Sources in a Bessel Cable with two Terminal conditions.

§ 332. Operational Solutions for Sources in the General Case.

§ 333. Conversion of Wire Waves to Cylindrical Waves. Two Ways.

§ 334. Special Cases of Zeroth Bessel Solutions.

§ 335. Numerical Interpretation of Formulæ. The Divergent Series.

§ 336. The divergent Formulæ are fundamental. Generation of Waves in a Medium whose Constants vary as the nth power of the distance.

§ 337. Construction of General Solution by Waves.

§ 338. Construction of General Solution by the Convergent Formulæ.

§ 339. Comparison of Wave and Vibrational Solutions to deduce Relation of Divergent to Convergent Bessel functions.

§ 340. Nature of Algebraical Transformation from Divergent to Convergent Formulæ.

§ 341. Rationality in p of Operational Solutions with two Boundaries. Solutions in terms of Im and Km.

§ 342. The convergent Oscillating Bessel functions, and Operational Solutions in terms thereof.

§ 343. The divergent Oscillating Bessel functions.

§ 344. Physical reason of the unlikeness of the two divergent functions Hm, Km.

§ 345. Electrical Argument showing the Impotency of Restraints at the origin, unless 1 >n> 1.

§ 346. Reduced Formulæ when one Boundary is at the Origin.

§ 347. The Expansion Theorem and Bessel Series. The Potential due to initial Charge.

§ 348. Time Function when Selfinduction is allowed for.

§ 349. The Potential due to initial Current.

§ 349A. Uniform Subsidence of Induction a.nd Displacement in combinations of Coils and Condensers.

§ 349 B. Uniform Subsidence of Mean Voltage in a. Bessel Circuit.

§ 349c. Uniform Subsidence of Mean Current in a Bessel Circuit.

APPENDIX C. RATIONAL UNITS.

CHAPTER VII. ELECTROMAGNETIC WAVES AND GENERALISED DIFFERENTIATION.

§ 350. Determination of the Value of pH by a Diffusion Problem.

§ 351. Elementary generalised differentiation. Value of pml when m is integral or midway between.

§ 352. Cable Problem:C=(K+Sp)½(R1+Lp)½V (1 ). Elementary Cases by Inspection.

§ 353. (2). Algebrisation when e is constant and K zero. Two ways. Convergent and divergent results.

§ 354. (3). Third way. Change of Operand.

§ 355. (4). V due to steady C whenR=O. Instantaneous Impedance and Admittance.

§ 356. (5). C due to steady V when S = O. The error function again.

§ 357. (6). V due to steady C when L=O. Two ways.

§ 358. (7). V due to steady C when S is zero. Two ways.

§ 359. (8). All constants finite. C due to V varying as Ept_

§ 360. (9). C due to V varying as ∈Kt/S, Discharge of a charged into an empty Cable.

§ 361. (10). C due to steady V, and V due to steady C.

§ 362. {11). C due to impulsive V, and V due to impulsive C.

§ 363. Cubic under radical. Reversibility of Operations. Distribution of Operators.

§ 364. (12). Development of Equation (60). C due to steady V.

§ 365. (13). Another Development of Equation (60).

§ 366. (14). Third Development of Equation (60). Integration by Parts.

§ 367. (15). Generalisation. The Complete Wave of C due to V at the origin varying as ∈Kt/S,

§ 368. (16). Summary of Work showing the Wave of C due to V at origin varying as ∈ Kt/s.

§ 369.(17). Derivation of the Wave of V from the Wave of C.

§ 370. (18). The Wave of V independently developed.

§ 371. (19). The Waves of V and C due to Initial Charge or Momentum at a Single Place.

§ 372. The Waves of V and C due to a Steady Voltage Impressed at the Origin.

§ 373. Analysis of Transmission Operator to Show the Deformation, Progression and Attenuation of Waves.

§ 374. Three Examples. The Wave of V due to impressed Voltage, varying as ∈  pt, ∈ Kt/S, or steady.

§ 375. Expansion of Distortion Operators in Powers of p1.

§ 376. Example. The Current Wave due to Impressed Voltage e0∈Kt/S at the Origin.

§ 377. Value of (p2  σ 2)*lo(σt) when n is a Positive or Negative Integer. Structure of the Convergent Bessel Functions.

§ 378. Analysis Founded upon the Division of the Instantaneous State into Positive and Negative Pure Waves.

§ 379. Simplest Solutions. Waves of Infinite Length, and of Length 2πv/σ.

§ 380. Development of GeneraI Solution in ump m and wmP m Functions.

§ 381. Derivation of C Wave from V Wave, and Conversely, with Examples. Condition at a Moving Boundary. Expansion Of ∈σ t .

§ 382. Deduction of the V and C Waves when VO is Constant from the Case VO = e∈  Kt/S. Expansion of ∈Pt in I Functions. Construction of the Wave of V due to any Impressed Voltage.

§ 383. Identical Expansions of Functions in In Functions. Formula for (½x)n. Electromagnetic Applications.

§ 384. Expansion of any Power Series in I,. Functions. Examples.

§ 385. Expansion of a Power Series in Jn Functions. Examples.

§ 386. The Waves of V and C due to an::v V0 developed in w"'Pm(z) Functions from the Operational Form of V 0∈pt.

§ 387. Remarkable Formula for the Expansion of a. Function in In Functions, and Examples. Modification of § 386, and Example.

§ 388 Impulsive Impressed Voltages, and the Impulsive Waves and their Tails generated.

§ 389. Tendency of Distortion to vanish in rapid Fluctuations. Effect of increased Resistance in rounding off corners and distorting.

§ 390. Wave Due to Impressed Voltage varying as ∈pt t1.

§ 391. Wave Due to V0 varying as ∈Ptlog t.

§ 392. Effect of a Terminal Resistance as expected in 1887 and as found in 1896.

§ 393. Reflection at the Free Ends of a Wire. A Series of Spherical Waves.

§ 394. Reflection at the End of a Circuit terminated by a Plane Resisting Sheet.

§ 395. Long Wave Formulie for Terminal Redaction.

§ 396. Reflection of Long Waves in General.

§ 397. Terminal Reflection without Loss. Wave Solutions.

§ 398. Comparison with Fourier Series. Solution of Definite Integrals.

§ 399. General Way of Finding Second and Following Waves due to Terminal Reflection.

§ 400. Application to Terminal Resistances. Full Solutions witb the Critical Resistances. Second Wave with any Resistance.

§ 401. Inversion of Operations. Derivation of First Wave from the Second.

§ 402. Derivation of Third and Later Waves from the Second.

§ 403. Summarised Complete Solutions.

§ 404. Reflection by a Condenser.

§ 405. Reflection by an Inductance Coil.

§ 406. Initial States. Expression of Results by Definite Integrals.

§ 407. The Special Initial States J0 (σx/v) and Jn (σx/v).

§ 408. The States of V and C resulting from any Initial States, V0 and C0, expansible in Jn functions.

§ 409. Some Fundamental Examples.

§ 410. The Genera.I Solution for any Initial State, and some Simple Examples.

§ 411. Conversion to Definite Integrals. Short Cut to Fourier's Theorem.

§ 412. The Space Integrals of V and C, due to Elements a.t the Origin.

§ 413. The Time Integral of C, due to Elements at the Origin.

§414. Evaluation of the Fundamental Integral.

§ 415. Generalisation of the Integral. Both kinds of Bessel Functions.

§ 416. The C due to initial V0 • Operational Method, and Modification.

§ 417. The V due to initial V0.

§ 418. Final Investigation of the V and C due to initial V0 and C0.

§ 419. Undistorted Waves without and with attenuation.

§ 420. Effects of Resistance and Leakance on an Initial State of Constant VO on one side of the Origin.

§ 421. Division of Charge Initially at the Origin into Two Waves with Positive or Negative Charge between them.

§ 422. The Current due to Initial Charge on one side of the Origin.

§ 423. The Aftereffects of an Initially Pure Wave. Positive and Negative Tails.

§ 424. Figs. 1 to 13 described in terms of Electromagnetic Waves in a doubly Conducting Medium.

CHAPTER VIII. GENERALISED DIFFERENTIATION AND DIVERGENT SERIES.

§ 425. A Formula for n obtained by Harmonic Analysis.

§426. Algebraical Construction of g(n). Value of g(n)g(  n).

§ 427. Generalisation of Exponential Function.

§ 428. Application of the Generalised Exponential to & Bessel Function, to the Binomial Theorem, and to Taylor's Theorem.

§ 429. Algebraical Connection of the Convergent and Divergent Series for the Zeroth Bessel Function.

§ 430. Limiting Form of Generalised Binomial Expansion when Index is 1.

§ 431. Remarks on the Operator (I+ Ll1)n.

§ 432. Remarks on the Use of Divergent Series.

§ 433. Logarithmic Formula derived from Binomial.

§ 434. Logarithmic Formulm derived from Generalised Exponential.

§ 435. Connections of the Zeroth Bessel Functions.

§ 436. Operational Properties of the Zeroth Bessel Functions.

§ 437. Remarks on Common and Generalized Mathematics.

§ 438. The Generalised Zeroth Bessel Function Analysed.

§ 439. Expression of the Divergent Ho{x) a.nd K0 (x) in Terms of Two Generalised Bessel Functions. Generalisation of ∈x.

§ 440. The Divergent Hn(x) and Kn(x) in Terms of Two Special Generalised Bessel Functions.

§ 441. The Divergent Hn(x) and Kn(x) in Terms of any Generalised Bessel Function of the same order.

§ 442. Product of the Series for ∈x and ∈x cos πr. Possible Transition from ∈x to ∈x.

§ 443. Power Series for log x.

§ 444. Examination of some Apparent Equivalences, and Rectification.

§ 445. Determination of the Meaning of a Generalised Bessel Function in terms of H0 and K0

§ 446. Some Apparent Equivalences.

§ 447. Cotangent Formula and Derived Formula for Logarithm. Various Properties of these and other Divergent Series.

§ 448. Three Electrical Examples of Equivalent Convergent and Divergent Series.

§ 449. Sketch of Theory of Algebrisation of (1 bp r)1 1.

APPENDICES.

APPENDIX D. ON COMPRESSIONAL ELECTRIC OR MAGNETIC WAVES.

APPENDIX E. DISPERSION.

APPENDIX F. ON THE TRANSFORMATION OF OPTICAL WAVE SURFACES BY HOMOGENEOUS STRAIN.*

1. Simplex Eolotropy.

2. Properties connected with Duplex Eolotropy.

3. Effects of straining a Duplex Wave Surface.

4. Forms of the Index and WaveSurface Equations, and theProperties of Inversion and Interchangeability of Operators.

5. General Transformation of WaveSurface by Homogeneous Strain.

6. Special Cases of Reduction to a Simplex WaveSurface.

7. Transformation from Duplex Wave to IndexSurface by aPure Strain.

8. Substitution of three successive Pure Straws for one .Two ways .

9. Transformation of Characteristic Equation by Strain.

10. Derivation of Index Equation from Characteristic.

APPENDIX G. NOTE ON THE MOTION OF A CHARGED BODY AT A SPEED EQUAL TO OR GREATER THAN THAT OF LIGHT.

APPENDIX H. NOTE ON ELECTRICAL WAVES IN SEA WATER.*

APPENDIX I. NOTE ON THE ATTENUATION OF HERTZIAN WAVES ALONG WIRES.*

NOTES.

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