Heaviside: Primer

Oliver Heaviside (1850-1925), a self-taught English telegraph engineer and mathematical physicist. This primer is built from the arbiter's final verdicts (arbiter.md), the three reviews (physicist.md, mathematician.md, contrarian.md) and my own reading of the three body pages in the folder (Electromagnetic Waves pp. 139, 142, 143). I re-derived the worked example myself; example.py in this folder reproduces every number. Updated 2026-10-05 with the first Investigator run (web research into Heaviside's own books and the archives) as weighed in the arbiter's section "Investigator findings (2026-10-05)". No verdict changed category. After the QE check of that update, material the first pass left out was restored (Heaviside's own 1887 evidence and caveat in 3.4; Arthur's observations and tube experiment, more archive records and the pension and membership episodes in section 7), matching the arbiter's additions.

Source tags: [read] = from the page images in the folder; [web] = from a web page, URL given; [web, via Investigator] = the Investigator's transcription of a web source (usually the archive.org machine-read text, OCR, of a scanned book), not rechecked by the arbiter unless said; [memory] = recalled, with confidence; [arbiter] = the arbiter's checked conclusion. Abbreviations: EP2 = Electrical Papers Vol. 2; EMT1, EMT2 = Electromagnetic Theory Vols. 1, 2; IET = Institution of Engineering and Technology (holds the Heaviside and Preece archives); IEE = Institution of Electrical Engineers (its predecessor); GPO = General Post Office.

Source gap. The folder holds only three saved Google Books viewer pages with nine page images: contents pages of Electrical Papers Vol. 1 and Electromagnetic Theory Vol. 1, and pp. 139, 142, 143 of a long Philosophical Magazine paper reprinted in Electromagnetic Waves. No full text of any work was available. Only claim 4 (waves, diffusion, thin-sheet reflection) rests on pages actually read. Everything about the four vector equations, the Poynting flux, the loading coil, the operational calculus, the ionosphere and his late positions rests on contents headings plus Wikipedia [web] and recall [memory]. To study Heaviside properly, obtain Electromagnetic Theory Vols. 1-3 (1893, 1899, 1912), Electrical Papers Vols. 1-2 (1892) and the 1888-89 Phil. Mag. papers; all are public domain [memory, high].

Update 2026-10-05. The Investigator has now read parts of Electrical Papers Vol. 2, Electromagnetic Theory Vols. 1-3 and the 1902 Encyclopaedia Britannica article "Telegraphy" in the OCR text on archive.org. Nothing was saved to the project, so every such quotation below is the Investigator's transcription [web, via Investigator]. The gap is partly closed: claims 3, 5, 6, 7a, 7b and 7c now have some of Heaviside's own words. But no formula for 7a or 7b, and none of eqs. (47)-(48), has been read from a page image (OCR garbles equations), and claims 1, 2 and 4 gain nothing. Claim 4 is still the only claim whose mathematics was read from page images [arbiter, Source gap].

---

1. The idea in one paragraph

Heaviside took James Clerk Maxwell's sprawling 1860s-70s theory of electricity and magnetism and rewrote it into the compact form every student now learns, then used it as an engineer's tool. He showed that energy travels through the space around wires, not inside them; that a telephone or telegraph line can be made to carry a signal without smearing it out, by adding inductance (the opposite of what the senior Post Office engineer William Preece insisted); that a sudden disturbance hitting a conductor is mostly reflected, and what gets in creeps in by "diffusion" that he pictured as countless small reflections; and that differential equations can be solved by treating the derivative d/dt as if it were an ordinary number. He did this outside any university, with no degree, publishing in trade journals. Three times the record shows his work being resisted: a Post Office official blocked a paper in 1887, the American Telephone and Telegraph Company (AT&T) hired two men in 1897 to find faults in his work, and the Royal Society rejected the third part of his paper on operators. The record also shows Heaviside provoking his opponents, and some senior figures backing him, so the story is not one-sided. Most of what he claimed was later shown to be right. Others took out the patents and were paid, while he refused money that did not come with "full recognition" and stayed poor. He also opposed Einstein's relativity. Relativity has since passed many tests, but his actual arguments against it have not been found or read for this review.

---

2. What you need to know first

Vector calculus and Maxwell's equations. You need divergence (how much a field spreads out from a point), curl (how much it circulates around a point) and the two integral theorems (Gauss, Stokes) that connect them to fluxes and loops. Maxwell's equations in vector form are exactly what Heaviside invented, so you are reading his notation whenever you open a modern text. Read: D. J. Griffiths, Introduction to Electrodynamics, chapter 1 (vector analysis) and chapter 7 (electrodynamics, Maxwell's equations) [memory, high].

Plane electromagnetic waves and wave impedance. In empty space a plane wave has an electric field E and a magnetic field H at right angles, both perpendicular to the direction of travel, in a fixed ratio E/H = Z0 ≈ 377 ohm (the "impedance of free space"). A wave going the other way has the same magnitude ratio but one field's sign flipped. Read: Griffiths chapter 9 (electromagnetic waves) [memory, high].

Boundary conditions at a surface. Where two regions meet, the component of E along the surface is continuous, and the component of H along the surface jumps by the surface current per unit width flowing in the boundary (Ampere's law applied to a thin loop straddling it). These two rules are all you need for the worked example. Read: Griffiths chapter 7 (boundary conditions for the fields), chapter 9 (reflection and transmission) [memory, high].

Energy flow: the Poynting theorem. Energy stored in the fields per unit volume is u = (1/2)(ε E² + μ H²); its rate of change plus the outflow of the vector S = E × H equals minus the work done on charges, J·E. Read: Griffiths chapter 8 (conservation laws) [memory, high].

Transmission lines. A pair of wires is modelled per unit length by four constants: series resistance R (ohm/m), series inductance L (H/m), shunt capacitance C (F/m) between the wires, and shunt leakage conductance G (S/m) through the insulation. These lead to the "telegrapher's equations" (Heaviside's own) and a complex propagation constant γ = α + jβ, where α is attenuation (nepers per metre; 1 neper = 8.686 decibels) and β is phase per metre. Read: D. M. Pozar, Microwave Engineering, chapter 2 (transmission line theory) [memory, high].

Complex exponentials (phasors). A sinusoid at angular frequency ω = 2πf is written as the real part of A e^{jωt}; then d/dt becomes multiplication by jω. (Engineers write j for the square root of -1.) This is the seed of Heaviside's operational calculus. Any ODE text covers it.

Laplace transform. Converts a linear ODE in t into an algebraic equation in a variable s (or Heaviside's p); solve algebraically, then transform back. Read: any ODE text, e.g. W. E. Boyce and R. C. DiPrima, Elementary Differential Equations, chapter on the Laplace transform [memory, high that it is covered; chapter number not given].

Old units. Heaviside in 1888 wrote in centimetre-gram-second electromagnetic units (CGS-EMU). You only need two translations: his μv for vacuum is the speed of light, 3 × 10¹⁰ cm/s, and the free-space impedance in his units is 4πμv (which equals 377 ohm once converted, since 1 ohm = 10⁹ EMU resistance units). His formulas carry factors of 4π that SI hides inside ε0 and μ0. (He later campaigned for "rationalised" units that remove these 4π factors; modern SI and Heaviside-Lorentz units both descend from that campaign [memory, medium].) Read: J. D. Jackson, Classical Electrodynamics, appendix on units [memory, high].

---

3. The textbook view

3.1 Maxwell's equations in Heaviside's vector form (claim 1)

In SI units, for fields in matter:

• ∇ · D = ρ (Gauss: electric flux leaves charge)
• ∇ · B = 0 (no magnetic charges)
• ∇ × E = -∂B/∂t (Faraday: changing B makes circulating E)
• ∇ × H = J + ∂D/∂t (Ampere-Maxwell: current and changing D make circulating H)

Symbols: E electric field (V/m); H magnetic field intensity (A/m); D = εE electric displacement (C/m²); B = μH magnetic flux density (tesla); ρ charge density (C/m³); J current density (A/m²), J = σE in a conductor of conductivity σ (S/m); ε permittivity, μ permeability; ∂/∂t time derivative.

Maxwell's own 1865 statement used about twenty equations in twenty unknowns, including the scalar and vector potentials. Heaviside (1884) eliminated the potentials and reduced twelve of them to these four in two unknowns, E and H [web] https://en.wikipedia.org/wiki/Oliver_Heaviside ; [arbiter, correction to "twenty replaced by four"]. Same predictions, far easier to use. (One caveat: in quantum mechanics the potentials turn out to have measurable effects, the Aharonov-Bohm effect [memory, high], so dropping them is a classical convenience.)

3.2 Energy flow (claim 2)

From the four equations one derives the Poynting theorem:

∂u/∂t + ∇ · S = -J · E, with S = E × H and u = (1/2)(E·D + H·B).

S (W/m²) is the energy flux. Poynting published it in 1884; Heaviside found it independently [web] https://en.wikipedia.org/wiki/Oliver_Heaviside. The integral form (total power through a closed surface) is tested every time anyone measures radiated or delivered power. The local form has a subtlety: you can add the curl of any vector field to S without changing ∇·S (since ∇·(∇×A) = 0 always), so "which way energy flows at this point" is not uniquely fixed by experiment [arbiter, from mathematician.md].

3.3 Waves in a conductor: wave plus diffusion (claim 4)

In a medium with conductivity σ the Maxwell equations give, for a plane wave along z,

∂²E/∂z² = μσ ∂E/∂t + με ∂²E/∂t².

The last term alone is the wave equation (speed v = 1/√(με)); the middle term alone is the diffusion (heat) equation. In a good metal at ordinary frequencies the diffusion term dominates, and fields soak in only to the skin depth δ = √(2/(ωμσ)), about 65 µm in copper at 1 MHz [memory, high]. Heaviside says the same on p.142: "In a good conductor ... the diffusion solution (50) applies practically. It is only in bad conductors that the more complete form is required" [read: page image p.142].

3.4 Transmission lines and the Heaviside condition (claim 3)

For a line with constants R, L, G, C per unit length, voltage V and current I obey the telegrapher's equations

∂V/∂z = -R I - L ∂I/∂t, ∂I/∂z = -G V - C ∂V/∂t.

For a sinusoid of angular frequency ω, waves vary as e^{-γz} with

γ = α + jβ = √((R + jωL)(G + jωC)).

Attenuation α generally depends on ω, and so does the speed ω/β. A voice signal is a mix of frequencies; if each is attenuated and delayed differently, the shape smears: this is distortion. Heaviside noticed that if

R/L = G/C (the Heaviside condition)

the square root factorises: γ = √(LC)·(R/L + jω), so α = √(RG) and the speed 1/√(LC) are both independent of frequency, and the line is distortionless [web] https://en.wikipedia.org/wiki/Heaviside_condition (cited in contrarian.md); [derivation checked by arbiter and in example.py]. For a real cable G is tiny, so R/L is far larger than G/C; the fix is to raise L. In the low-loss approximation α ≈ R/(2Z0) + G Z0/2 with Z0 = √(L/C); the arbiter checked numerically that this is smallest exactly at the Heaviside condition, so added inductance helps up to that balance and hurts past it [arbiter, dispute 7]. According to the Loading coil page, Heaviside found this condition in 1887 "By applying his operational calculus" to the line [web] https://en.wikipedia.org/wiki/Loading_coil . So his best-known engineering success was also an early use of the method in 3.5, the method whose rigour was disputed.

His own 1887 words. The condition was in print, in Heaviside's name, in the trade weekly The Electrician in 1887, twelve years before the first test. The Investigator found it reprinted in Electrical Papers Vol. 2, Section XL (pp. ~119-124) [web, via Investigator: https://archive.org/details/electricalpapers02heavrich]. He wrote "Make R/L = K/S, and the distortion is annihilated" (his K is leakage conductance and S capacitance per unit length, so this is R/L = G/C). He gave the exact solution, a wave V = e^(-Rt/L) f(x - vt) that keeps its shape f while shrinking, and wrote: "The energy is half electric, half magnetic; the dissipation is half in the wire, half [in the leak]". He also said the older Kelvin cable theory "wholly ignores inertia" (that is, inductance). You can check the solution yourself. Combining the telegrapher's equations gives

∂²V/∂x² = LC ∂²V/∂t² + (RC + GL) ∂V/∂t + RG V.

Substitute V = e^(-at) f(x - vt) with a = R/L and v = 1/√(LC). The f'' terms cancel because LCv² = 1. The f terms cancel because LCa² - (RC + GL)a + RG = R²C/L - R²C/L - GR + RG = 0. What is left is v(RC - GL) f'(x - vt), which vanishes for every pulse shape f exactly when GL = RC, which is R/L = G/C. All three reviewers re-derived this, and the mathematician found numerically that attenuation is smallest at L = 4.9996 mH/km against RC/G = 5 mH/km [arbiter, claim 3 and "My own checks" (c)]. (Kelvin's 1855 cable formula is the limit L = 0, G = 0 of the same equation: correct for slow cable telegraphy, wrong for voice frequencies, where ωL exceeds R [arbiter, claim 3].)

What the 1887 section does not fix is the date of the proposal to use coils. The historian Bruce Hunt (Physics Today, 2012) places coils at intervals in the 1886-87 work; Wikipedia and Grace's Guide give 1893; the 1887 section itself raises inductance by wire spacing and iron, not coils. The coil date is unsettled between 1887 and 1893 [arbiter, claim 3].

His own evidence, and his own warning. The same section (EP2 Sec. XL, p. 122 [web, via Investigator: https://archive.org/details/electricalpapers02heavrich]) carries one piece of evidence for the idea and one caveat against overdoing it, both in his words [arbiter, claim 3]. - For. He cites reports of good long-distance telephony on copper-covered steel wires, where current entering the steel "lowers the resistance and increases the inductance", and offers it cautiously: "I do not wish to magnify its importance". This is his report of other people's practice, not a controlled test of his own (mathematician, contrarian), and it raises L with iron, not with coils. - A limit he flagged. Pushed to an impracticable extent, the method "could make the amplitude of sinusoidal currents received at the distant end of an Atlantic cable greater than the greatest possible steady current from the same impressed force -- an unbelievable result". He blames it on waves reflected to and fro along the line ("This is to be avoided"). The reviewers read it the same way: the physicist calls it plausible as a resonance effect, not verified; the mathematician calls it a standing-wave resonance on a mismatched line of finite length, "mathematically ordinary, not a paradox"; the contrarian calls it "a candid caveat, not an error I can find". The arbiter agrees. The idea in plain terms: a line of finite length that is not terminated in Z0 reflects waves back and forth, like a pipe that resonates at certain notes. At a resonant frequency the reflections add in step, and the received amplitude at that one frequency can exceed the direct-current (DC) value, just as a driven, lightly damped oscillator can swing wider than its static deflection. So the statement is not self-contradictory. It is a warning against over-loading, and it fits the "critical value of the inductance" in the same section and the minimum in α described above. Nobody has computed his specific Atlantic-cable case.

3.5 Operational calculus (claim 5)

Write p for d/dt. Example: solve dy/dt + a y = 1 for t > 0 with y(0) = 0. Heaviside writes (p + a) y = 1, so y = [1/(p + a)]·1, and his rules (the "expansion theorem") turn this into y = (1/a)(1 - e^{-at}), which is correct. Today the same steps are justified by the Laplace transform: transform, solve algebraically, invert. Bromwich supplied a rigorous contour-integral justification after Heaviside [web] https://en.wikipedia.org/wiki/Oliver_Heaviside. The folder shows the operator in use only once: Electromagnetic Waves p.142, sec. 10 "Effect of Impressed Force", says "In equations (23), (24), let p = ni, where n/2π = frequency of alternations" [read: page image p.142]. Here n is the angular frequency (ω) and i is the square root of -1 (his i, the engineer's j). That is exactly the phasor rule of section 2: for sinusoidal driving, d/dt becomes multiplication by iω. The expansion theorem and the harder cases (divergent series) are not in the folder [source gap] [arbiter, QE revision].

How the method was resisted. Heaviside sent "On Operators in Physical Mathematics" to the Royal Society in 1892-93. Parts I and II were printed (Part I: [web] https://gwern.net/doc/math/1892-heaviside.pdf ; Part II: [web] https://archive.org/details/philtrans01936980 ). Part III was refereed and rejected. Hunt (Physics Today, 2012), read verbatim by the Investigator: "pure mathematicians objected to the cavalier way he handled divergent series [infinite sums that have no finite total], and when he submitted a third installment, it was sent to a referee and rejected" [web, via Investigator: https://physicstoday.aip.org/features/oliver-heaviside-a-first-rate-oddity]. A Google search snippet of Hunt's 1991 book chapter names the referee as the mathematician William Burnside (Royal Naval College, Greenwich); the chapter and the referee's report have not been opened, so the name rests on that snippet alone. Heaviside answered in print, in Electromagnetic Theory Vol. 2 (1899), sec. 425: "if investigations of this matter are not good enough for Another Place, how can they be good enough for Electromagnetic Theory? ... the wet blankets of rigorists"; and on p. 9: "Shall I refuse my dinner because I do not fully understand the process of digestion? No, not if I am satisfied with the result." [web, via Investigator: https://archive.org/details/electromagnetict02heavrich]. A secondary source (deadreckonings.com, snippet only) says Vol. 2 contains a modified Part III. Each of these is that party's own stated position. The referee objected to the argument, not to wrong results; Heaviside's reply is a test by outcome, which works until it does not. No source read in this review shows a case where his method gave a wrong answer [arbiter, claim 5].

3.6 The ionosphere (claim 6)

An ionised gas reflects radio waves below its plasma frequency, f_p ≈ 8.98 √n hertz, with n the electron density per cubic metre; n = 10¹¹ m⁻³ gives about 2.8 MHz [arbiter, recomputed]. Kennelly and Heaviside independently proposed a conducting layer in 1902; Appleton showed it in 1924 (Nobel Prize 1947), Breit and Tuve measured its height in 1925 [web] https://en.wikipedia.org/wiki/Kennelly%E2%80%93Heaviside_layer. (The Heaviside Wikipedia page says 1923. Two more sources found by the Investigator give 1924: Bruton 2018, who dates Appleton's experiments from late 1924, and the IET archives blog [web, via Investigator: https://royalsocietypublishing.org/rsta/article/376/2134/20170454/115714/From-theory-to-engineering-practice-shared ; https://ietarchivesblog.org/2023/04/05/from-under-the-sea-to-the-edge-of-space-the-work-of-oliver-heaviside/]. The "1923" is unexplained [arbiter, dispute 5].)

What Heaviside actually wrote. His 1902 text is in the Encyclopaedia Britannica, 10th edition, vol. XXXIII, "Telegraphy", I. Theory (pp. ~213-215): "There may possibly be a sufficiently conducting layer in the upper air. If so, the waves will, so to speak, catch on to it more or less. Then the guidance will be by the sea on one side and the upper layer on the other." [web, via Investigator: https://archive.org/details/in.ernet.dli.2015.209824]. (The author's initials are garbled in the OCR; most likely Heaviside, not confirmed.) Notice three things. It is tentative ("may possibly"). It describes guidance between two conductors, like a wave in the gap between two metal plates, not reflection from a plasma. And on the citation dates it came about nine months after Kennelly's article (Electrical World and Engineer, 15 March 1902), which nobody in this review has read. The physicist notes that guidance between the sea and a conducting layer is physically sound for very-low-frequency waves (the Earth-ionosphere cavity) [memory, high], while long-range short-wave radio works by refraction and reflection from the layer's electron-density gradients [memory, high]. So the layer is Established; Heaviside's sentence is a priority question rather than a separate testable claim: tentative, qualitatively right for guidance, silent on reflection, second by publication date, credit shared, as the name Kennelly-Heaviside layer already says. The contrarian had written that "the reflection form is Kennelly's"; the arbiter set that aside because it assumes a content for Kennelly's paper that no one has read [arbiter, claim 6].

---

4. Where Heaviside departs (and where he was the textbook)

Heaviside is unusual for this collection: most of his "departures" from the physics of his day are now the textbook. The table puts his position next to the view he opposed, then next to today's view.

4.1 The page actually read: the thin metal sheet

This is the one place we can watch Heaviside work. On p.143, section 11, "True Nature of Diffusion in Conductors", he writes that diffusion of magnetic induction into a conductor "appears to be fundamentally one of repeated internal reflexions with partial transmission" [read: page image p.143]. To illustrate, he sends a plane wave E1 = μvH1 at an "exceedingly thin sheet of metal" of conductivity k1 and thickness z. With transmitted wave E2 = μvH2 and reflected wave E3 = -μvH3 he writes

E1 + E3 = E2 (60) H1 + H3 = H2 + 4π k1 z E2 (61)

and concludes

E2/E1 = H2/H1 = 1/(1 + 2π μ k1 z v) (62)

"H is reflected positively and E negatively. A perfect conducting barrier is a perfect reflector, it doubles the magnetic force and destroys the electric force on the side containing the incident wave, and transmits nothing." For copper, k1 = 1/1600 and μv = 3 × 10¹⁰, half transmission needs z = (2πμk1v)⁻¹ = (8/3π) × 10⁻⁸ cm (63) [read: page image p.143].

Side by side with the textbook: (60) is "tangential E is continuous"; (61) is "tangential H jumps by 4π × surface current" (CGS-EMU), where the surface current per unit width is conductivity × thickness × E, i.e. k1 z E2. The modern SI result is E2/E1 = 1/(1 + Z0 σ d/2). The two are identical once you note that 4πμv is Z0 in his units. There is no departure from the textbook here; this is the textbook, written in 1888 [arbiter: Established for eqs. 60-62].

Where he goes beyond what the pages show: - Eqs. (47)-(48) on p.139 (Bessel-function solutions for a pulse in a leaky medium) are asserted without proof: "I forbear from giving a detailed demonstration ... although they were very laboriously worked out by myself" [read: p.139]. The form is right for this type of equation; the prefactors were not checked by anyone in this review because his definitions of s, q and a are on pages not in the folder [arbiter: Plausible, untested; the check still to do is a calculation]. - On p.139 he states that a medium with both electric conductivity k and a (fictitious) magnetic conductivity g "of the right amounts" produces no tail behind a travelling pulse [read: p.139, sentence runs onto the missing p.140]. This is the field version of his distortionless line: a magnetic conductivity plays the role that series resistance R plays on a line [memory, medium, that this is the origin of the line condition]. P.143 supports this reading with a duality statement: "To produce dissipation of H with persistence of E requires the plate to be a magnetic, not an electric conductor", and "the vorticity of e produces magnetic induction, that of h produces electric displacement" [read: page image p.143]. In words: an electric conductor (conductivity k) eats E, a magnetic conductor (g) would eat H, and the two roles mirror each other when E and H swap. Balancing k against g makes the losses of E and H match, so the pulse keeps its shape and only shrinks [arbiter, QE revision: supports claim 4, verdict unchanged]. - On p.142 he also gives the high-frequency limit: as n (angular frequency) grows, k and g act only as small attenuation exponents and E and H come into phase [read: page image p.142]. This is the low-loss limit in the textbook: when ωε ≫ σ the wave travels almost as in vacuum and only decays slowly. Compare 5.6: at high frequency (ωL ≫ R) the line's α approaches R/(2Z0) + G Z0/2, independent of frequency. - The sentence after (63), "The H disturbance is made 3/2 H1, the E reduced to 1/2 E1 on the transmission side" [read: p.143]: the total H of 3/2 H1 is on the incident side (H1 + H3 = H1 + H1/2); on the transmission side H2 = H1/2. The sentence can be read two ways. If "on the transmission side" applies to both clauses, the H clause names the wrong side. If it applies only to the E clause, the sentence is exact: H is 3/2 H1 at the incident face, and E is 1/2 E1 on both sides because E is continuous. The printed text does not decide between the readings, so the arbiter records it as ambiguous phrasing, not as an error by Heaviside [arbiter, dispute 1, revised 2026-10-05]. Either way, the equations are correct.

4.2 Two late departures, now read in transcription (claims 7a and 7b)

These passages were read only as OCR text, where the equations are garbled, so you can follow his words but not yet his formulas.

The force inside a magnet (7a; Electromagnetic Theory Vol. 1, sec. 83, pp. 103-104, heading "Difficulty. Maxwell's Solution probably wrong"). Heaviside accepts that the resultant (total) force on a magnet can be found from the stress on a surface outside it. But "it is impossible that the real forcive can be represented merely by the surface traction"; to get it "we must go inside the magnet. This can only be done hypothetically, and with precarious validity". Maxwell's way "I believe to be quite erroneous for many reasons, the principal being that it does not harmonise with his scheme generally, and that it lumps together intrinsic and induced magnetisation, which have essential differences and are physically distinct." [web, via Investigator: https://archive.org/details/electromagnetict01heavuoft]. In modern terms: write the magnetisation (magnetic moment per unit volume) as M = M0 + χH, where M0 is the permanent ("intrinsic") part and χH the part induced by the field H, with χ the susceptibility. Different force-density formulas (Kelvin's, Helmholtz's, Maxwell's stress) all give the same total force on a body but can differ in how the force is spread inside it, and the difference sits in the M0 part [arbiter, from mathematician.md; memory, medium for the force-density point]. So Heaviside is not disputing anything a balance measuring the pull on a magnet could detect. He disputes the internal distribution.

The reviewers split on the label. The physicist and mathematician said "Not testable as stated": total-force measurements cannot separate the formulas, and his own text calls the internal route hypothetical. The contrarian said "Never properly tested". The arbiter kept Never properly tested: the usual measurements cannot decide it, but a measurement of internal stress or strain in a permanent magnet could in principle (in real materials it is confounded by magnetostriction, the change of shape a material undergoes when magnetised), and no such measurement is on record [arbiter, claim 7a]. His own formulas (pp. 103-107) still need page images.

Gravity as a field like electromagnetism (7b; Electromagnetic Theory Vol. 1, Appendix B, from The Electrician, 1893, pp. 455-466). Heaviside lets gravity spread through the ether at a finite speed v. The arbiter read Part I of the article in the OCR directly [web: https://archive.org/download/electromagnetict01heavuoft/electromagnetict01heavuoft_djvu.txt]. Moving matter, ρu (density ρ times velocity u), is treated as "analogous to a convective current of electrification"; a field h circles a moving particle along lines of latitude, as a magnetic field circles a moving charge; and the energy flux, the gravity version of E × H, comes out reversed, "since the radial e is directed to instead of from the centre ... This reversal arises from all matter being alike and attractive, whereas like electrifications repel one another." So the structure that today is called gravitoelectromagnetism is in his text. What the arbiter did not find in the passages read is a calculation for a spinning body (the "frame-dragging" of gyroscopes that Gravity Probe B measured), and the coefficients are unchecked because the equations are garbled [arbiter, claim 7b].

His prediction for a moving body: "no aberrational effect, but only a slight alteration in the intensity of force", weaker along the line of motion and stronger at right angles to it, "the direction is still radial" [web, via Investigator]. The size of the change goes as (u/v)², the square of the body's speed over the speed of gravity, which he takes as the speed of light, v = 3 × 10¹⁰ cm/s. Here is a numerical check worth doing yourself. The Investigator's note gave u = 3 × 10⁶ cm/s and (u/v)² = 10⁻⁶. But (3 × 10⁶ / 3 × 10¹⁰)² = (10⁻⁴)² = 10⁻⁸, not 10⁻⁶. All three reviewers caught this. The arbiter fetched the OCR directly: it reads "u = 3 x 10^ centim. per sec." with the exponent missing, so "10⁶" was never in the text. The OCR text of Heaviside's words says that u "is not very different from the speed attributed to fast stars", that the ratio squared is "one-millionth", and that the Sun-Earth attraction rises by "one two-millionth part". These fit u = 3 × 10⁷ cm/s = 300 km/s: (3 × 10⁷ / 3 × 10¹⁰)² = (10⁻³)² = 10⁻⁶. So Heaviside's text is consistent; the slip was in the transcription, and a page image would confirm the printed exponent [arbiter, claim 7b]. (The "two-millionth" implies a factor 1/2 multiplying (u/v)² in his formula; the formula itself is unread, so this is arithmetic on his words, not a check of his equation.)

Sec. 68 of the same volume names a "gravitational difficulty": "we are entirely ignorant of its distribution in the ether, and of its mode of transference" [web, via Investigator]. That is, nobody knew where gravitational energy sits or how it flows, unlike E × H for electromagnetism. It is not a flaw in the Poynting flux [arbiter, claim 7b].

---

5. Worked example: Heaviside's thin copper sheet

Chosen because it is the one calculation we can read in his own hand, every step uses only first-year E&M, and it hides a modern result (maximum absorption of a thin film) that Heaviside's numbers already contain. A short line calculation for the Heaviside condition follows in 5.6. Run example.py to reproduce everything:

.venv/bin/python review/Heaviside/example.py

5.1 Set-up (SI)

A plane wave in air, amplitude E1, hits a metal sheet of conductivity σ and thickness d, much thinner than the skin depth (so the field is uniform across the sheet). In air, H = E/Z0 for a wave going forward, H = -E/Z0 for one going backward, Z0 = 376.73 ohm. Unknowns: transmitted E2, reflected E3.

5.2 Boundary conditions

• Tangential E continuous: E1 + E3 = E2.
• Tangential H jumps by the surface current K = σ d E2 (A/m): (E1 - E3)/Z0 = E2/Z0 + σ d E2.

Multiply the second by Z0: E1 - E3 = E2 (1 + Z0 σ d). Add the first: 2 E1 = E2 (2 + Z0 σ d). So

E2/E1 = 1/(1 + x), x = Z0 σ d / 2, and E3/E1 = E2/E1 - 1 = -x/(1 + x).

In Heaviside's units x = 2π μ k1 z v. This is his eq. (62). Perfect conductor (σ → ∞): E2 → 0, E3 → -E1, so E on the incident side is E1 + E3 = 0 and H there is H1 + H3 = 2H1. That is his "doubles the magnetic force and destroys the electric force" (example.py, A1).

5.3 Heaviside's number

Half transmission means x = 1.

His units: 2π × (3 × 10¹⁰ cm/s) × (1/1600) = 1.178 × 10⁸ per cm, so z = 1/(1.178 × 10⁸) = 8.488 × 10⁻⁹ cm. His printed (8/3π) × 10⁻⁸ = 8.488 × 10⁻⁹ cm. Exact match (A2).

SI: d = 2/(Z0 σ).

The 5-8% spread is only his rounded copper conductivity [arbiter, dispute 2].

Units check of x: Z0 (ohm) × σ (S/m = 1/(ohm m)) × d (m) is dimensionless. Good.

Physical reality check. 0.085 nm is about a third of the spacing between copper atoms (about 0.25 nm [memory, high]). A copper film that thin cannot exist, and real films a few nanometres thick have higher resistivity than bulk [memory, medium]. So (63) is a correct formula applied to an impossible sheet. Heaviside presents it as a calculation for a thin sheet, not as a claim about real foil; his own comparison ("a very small fraction of the wave-length of visible light") points the same way [arbiter]. The useful form is the sheet resistance R_s = 1/(σ d): half transmission needs R_s = Z0/2 = 188 ohm per square, which is easy to make from thin resistive films (A3).

5.4 Fields on each side at half transmission

E2 = E1/2, E3 = -E1/2, so H3 = +H1/2 (reflected H keeps its sign, "H is reflected positively"). - Incident side: E = E1 + E3 = E1/2; H = H1 + H3 = 3/2 H1. - Transmitted side: E = E1/2; H = H1/2.

Check (61): jump in H = 3/2 H1 - 1/2 H1 = H1, and the sheet current gives σ d E2 × Z0 (in units of H1 = E1/Z0) = 2x × 1/2 = 1 at x = 1. Consistent. This is where "3/2 H1" comes from (A4).

5.5 Energy audit

Intensity is proportional to E² in the same medium, so - transmitted T = (1/(1+x))², at x = 1: 1/4 - reflected R = (x/(1+x))², at x = 1: 1/4 - absorbed A = Joule heat σ d E2² divided by incident intensity E1²/Z0 = 2x/(1+x)², at x = 1: 1/2.

T + R + A = 1 for every x (SymPy confirms identically). And dA/dx = 0 at x = 1: Heaviside's half-transmission sheet is also the sheet that absorbs the most, and no single thin sheet in free space can absorb more than half of an incident wave. This is a standard result in modern thin-film absorber design [memory, high]; it falls straight out of his 1888 equations. His remark "there is persistence of H, although there is dissipation of E" reflects that at the sheet the energy lost is drawn from the electric field, which drives the current, while the magnetic field builds up on the incident side.

5.6 Short companion: does added inductance really help?

Illustrative cable (my choice of typical values, not from any Heaviside document): R = 50 ohm/km, C = 50 nF/km, G = 1 µS/km. The Heaviside condition needs L = RC/G = 2.5 H/km. Exact α from γ = √((R + jωL)(G + jωC)) (example.py Part B):

(α in nepers per km.) Hand check for the loaded line: Z0 = √(0.044/5 × 10⁻⁸) = 938 ohm, α ≈ R/(2Z0) + G Z0/2 = 0.0267 + 0.0005 = 0.0271. Bare pair: ωL = 3.8 ohm/km is tiny next to R, and α ≈ √(ωRC/2) = 0.089, close to the exact 0.0855. The bare line loses 37 dB over 50 km and treats 3 kHz almost three times as harshly as 300 Hz; adding 44 mH/km (about the level real loading coils gave [memory, medium]) cuts the loss to under 12 dB and nearly removes the distortion. Preece's intuition ("inductance impedes current") was right about a single coil in a circuit and wrong about a line, where inductance and capacitance together set the wave impedance.

---

6. Evidence table

Why 7a and 7c get different verdicts. For 7a we know what he claimed, and since 2026-10-05 we have his argument in his own words (4.2): Maxwell's treatment of the internal force on permanently magnetised matter is wrong because it merges permanent and induced magnetisation. That is a definite statement about something physical (stress inside a magnet), so it is testable in principle, even though the usual measurement, total force, cannot decide it. The arbiter changed the label from "Not testable as stated" to "Never properly tested" earlier the same day and kept it after reading the argument; the physicist and mathematician still prefer "Not testable as stated", and both views are on record [arbiter, claim 7a]. For 7c we do not even know what he claimed. "Opposition" is not a statement anyone can test until his actual objection is read. It is not in Electromagnetic Theory Vol. 3 by keyword ("Einstein" 0 times, "Minkowski" 0, "relativ" 21, "Lorentz" 30 [web, via Investigator: https://archive.org/details/electromagnetict03heavuoft]); the arbiter draws no conclusion from a keyword count, since OCR can hide a name and Vol. 3 largely collects earlier work. It may be in the unpublished Vol. 4 notes or his 1903-1905 letters to Nature. Two records of his relation to relativity's ancestry are kept without weighing: Hunt says Föppl's 1894 textbook carried Heaviside's form of the equations, "from which Albert Einstein, among others, would learn electromagnetic theory"; and Russell (Nature 1925, an admirer) says he "was one of the first to predict the increase of mass of a moving charge" [web, via Investigator]. The three reviewers differed on 7c, and all three views stay on record: the physicist called it "Not assessable from the folder"; the mathematician "Historically superseded but instructive"; the contrarian "Not testable as stated", with "Tested, not supported" as far as the record shows [arbiter, claim 7c].

---

7. What a learner should take away

Solid. The four vector equations, the Poynting theorem, the telegrapher's equations, the distortionless condition, and the thin-sheet result are all correct and in daily use. The worked example shows his 1888 numbers match a modern SI calculation to the last digit once units are translated.

Wrong or unproven, and why it is instructive. - The "3/2 H1 on the transmission side" sentence: the equations are right, and the sentence is exact if "on the transmission side" applies only to E, but it is easy to read it as putting the H field on the wrong side. Lesson: always check a verbal summary against the equations, yours included. - Eq. (63) gives a thickness smaller than an atom. Heaviside presents it as a calculation for a sheet, not as a claim about real foil, but the lesson for you is that a correct formula can be pushed outside its domain (bulk conductivity, continuous metal). Always ask whether the numbers make physical sense. - Eqs. (47)-(48) were asserted without proof, and the operational calculus was used before it was justified. "It works" is not a proof, which was the Royal Society referee's objection. Heaviside's reply was that right answers "ought to be justification enough". No source read here names a case where his method gave a wrong answer; whether any did is an open question below. - His opposition to relativity: his arguments are unread, so this review cannot say where he went wrong, only that relativity has since passed many tests [memory, high]. Being right against authority once is not evidence of being right the next time, and being wrong once would not undo his earlier work either. - Wikipedia's "confirmed by Gravity Probe B" for his gravity analogy: GP-B's published comparison is with general relativity, not with his 1893 equations, and no source compares the two. The sources show neither confirmation nor refutation of his form. His 1893 text does contain the structure (a magnetic-like field round moving mass, 4.2), but no frame-dragging calculation was found and his coefficients are unchecked. An analogy that resembles a later tested theory has not itself been tested until someone does the comparison (see the open questions). - The power-of-ten slip in 4.2 is a lesson about transcriptions, not about Heaviside: a missing exponent in machine-read text became "10⁶" in a note, and the error showed up only when someone checked (u/v)² against the printed "one-millionth". Recompute any number you are handed before you build on it. - His 1902 ionosphere sentence: the claim as usually worded ("explains long-distance radio") is stronger than what he wrote, a tentative guess about guidance. Read the primary text before crediting or blaming anyone.

The institutional story, stated precisely. The distortionless condition was published by Heaviside in The Electrician in 1887 (3.4). What was dismissed without testing in 1887 was the joint paper by Oliver Heaviside and his brother Arthur, a Post Office engineer [memory, high: that Arthur was his brother], which Preece blocked; the Electrician series then stopped. The idea was tested and confirmed in 1899 by an industrial lab [web] https://en.wikipedia.org/wiki/Oliver_Heaviside ; https://en.wikipedia.org/wiki/Loading_coil .

How the block happened is reported two ways, and the arbiter does not choose between them [web, via Investigator; arbiter, claim 3]: - Before submission. Hunt (Physics Today 2012, historian) and Elizabeth Bruton (Phil. Trans. R. Soc. A 2018, Science Museum curator, citing IET archives): the April 1887 paper for the Journal of the Society of Telegraph Engineers needed clearance from Arthur's superior, Preece, who "promptly declared the paper worthless and blocked it". - Rejected by the Society. The 1925 obituary by the Institution of Electrical Engineers (IEE), written by an admirer (on Grace's Guide): Preece's circle "caused his papers and those of his brother ... to be rejected by the Society of Telegraph Engineers". - A second blocked item. Bruton, from an IET notebook (SC MSS 005/I/1/12): Parts VIII and IX(?) of "On the Self Induction of Wires" were "declined by the Philosophical Magazine and blocked by Preece, and partly published in The Electrician". - Heaviside's own view. In Electromagnetic Theory Vol. 1 sec. 1 he gives the stated reason for the end of the Electrician series, "pressure on space and the want of readers", then adds "Perhaps there were other reasons ... We do not dwell in the Palace of Truth". A reply written 24 Sept 1887 but "now first published" in 1892 (EP2 Art. 38) says "Mr. Preece is wrong, not merely in some points of detail, but radically wrong". - The editor. Hunt, citing Basil Mahon's 2009 biography: the Electrician's editor Biggs, a supporter, declined Heaviside's anti-Preece letters for fear of libel and was "abruptly removed" in October 1887, "a move he later hinted was prompted by his support for Heaviside". This is Biggs's own hint, in one secondary source; no second source has been found yet.

The records that would decide between "before submission" and "rejected" are the Society's 1887 council minutes (a paper never submitted leaves no council record) and the IET notebook itself; neither has been seen.

Several parties had a stake, and each one's account is given here as that party's own, next to its stake (Wikipedia items [web], same two pages, re-checked by the arbiter 2026-10-05; archive items [web, via Investigator]):

• William Preece was a senior Post Office engineer and Arthur Heaviside's superior. He had publicly taken the opposite view: "Preece had recently declared self-inductance to be the great enemy of clear transmission." He blocked the paper "because part of the proposal was that loading coils (inductors) should be added". The Heaviside article states that "Preece's motivations in suppressing Heaviside's work were more to do with protecting Preece's own reputation and avoiding having to admit error than any perceived faults in Heaviside's work". That is the article's characterization; the primary evidence behind it is not cited in what the arbiter read. Weigh it against the documented facts: Preece had committed himself publicly to the opposite view, and he had the power to block the paper. The same article quotes an unnamed author calling Preece "a powerful government official, enormously ambitious, and in some remarkable ways, an utter blockhead". That is a hostile characterization and should be read as one. The Post Office never took up the idea: Heaviside "never succeeded in persuading the British GPO to take up the idea". The Post Office record is one of not adopting the idea, not of a test that failed. Preece's own words are now glimpsed in one record: an IET catalogue description (the letter itself was not read) of his pencil draft reply to Kelvin (SC MSS 022/VII/526) quotes "Heaviside and his dreadful twaddle", "calculating Kelvin's formula to show that he is correct and Heaviside is wrong", and "drawing room philosophers like Oliver Heaviside and Sylvanus Thompson". This fits two readings, and both may be true at once: a stake (a public position, a costly iron-to-copper wire change justified by it, seniority over Arthur), and a model that left out inductance. Kelvin's formula is the L = 0, G = 0 limit of the telegrapher's equation (3.4), right for slow cable telegraphy and wrong for voice, so a man reasoning from it could reach Preece's conclusion in good faith. The second reading rests on a catalogue summary only [arbiter, claim 3].
• AT&T stood to profit from the technique and later owned the patents. "In 1897, AT&T employed one of its own scientists, George A. Campbell, and an external investigator Michael I. Pupin to find some respect in which Heaviside's work was incomplete or incorrect." Its engineers "estimated that they stood to save $700,000 in new installation costs in New York and New Jersey alone", and it is estimated AT&T "saved $100 million in the first quarter of the 20th century". AT&T bought an option on Pupin's patent "Fearing that there was a risk that the battle would end with the invention being declared unpatentable due to Heaviside's prior publication". In other words, a commercial party's own reasoning recognised that Heaviside published first.
• Michael Pupin, the patentee, was paid a large sum, reported as $455,000 in total (Wikipedia), "over $400 000" (IET archives blog) or $500,000 (Hunt 2012, who calls the circumstances "somewhat shady"); no AT&T/Pupin contract has been seen, so the true total is unsettled [arbiter, dispute 8]. The mathematician Norbert Wiener later denounced AT&T and Pupin "as thieves" (in The Tempter, 1959, per Hunt; a novel [memory, high]): Wiener's characterization. "Pupin himself claims that he first thought of the idea of loading coils while climbing a mountain in 1894, although there is nothing from him published at that time." That is Pupin's own account, given by a man with a large financial stake. Heaviside's proposals of 1887 and 1893 come before it [arbiter, claim 3].
• Heaviside had no patent on loading and received no payment. "Heaviside never patented his idea"; AT&T "offered Heaviside money in exchange for his rights", and he refused, "declining to accept any money unless the company were to give him full recognition"; "Heaviside was chronically poor".

Is this a case of authority deciding before experiment? Yes, and also a case where experiment, once run, settled the question quickly. It was not the only episode of resistance on record. The others were AT&T's 1897 commission to find fault with his work, and the Royal Society referee's rejection of Operators Part III (3.5). In the first two, the resisting party had a stake. In all three, the work was later borne out in use [arbiter, contrarian-limit paragraph]. The contrarian's own caution still applies: this is a vindication story, not evidence that outsiders are generally right, and his vindicated claims were extensions of established physics with clear, cheap tests.

The new records also apply the same rigour to Heaviside, and they make the story less one-sided [arbiter, contrarian-limit paragraph and claim 3; all web, via Investigator]: - Provocation on his side. In 1873 his paper on duplex telegraphy disparaged the work of Culley and Preece and did not cite Preece's 1855 duplex patent. Preece wrote to Culley that it was "most pretentious and impudent ... He must be met somehow"; Culley replied "We will try to pot Oliver somehow" (Bruton, citing an archive; letters not seen). So the conflict, and Preece's hostility, predate loading by fourteen years. - Kelvin. On 12 July 1887 Kelvin wrote to Preece that a Heaviside footnote in the Phil. Mag. "should not have appeared as it was so ill natured" (IET catalogue wording, SC MSS 022/VII/535). Yet Kelvin praised his propagation theory in his 1889 address to the IEE and helped it into the Phil. Mag. (Hunt). The resistance was not uniform. - Support. George FitzGerald wrote in his notebook on 9 Sept 1888: "Very important ... Must be all in O. Heaviside" (Trinity College Dublin, per Hunt), a contemporary endorsement. The 1925 obituaries (IEE; Russell in Nature, both admirers) name Silvanus Thompson and Pupin as early adopters. - Arthur's observations and experiment. Oliver reports in EP2 that Arthur's observations of oscillations on telegraph lines, which made the pointers of ABC telegraph indicators jump, prompted part of the induction work (pp. ~120s), and that Arthur's experiment with a tube as the return conductor found no inductive action outside the tube (pp. ~185-186) [web, via Investigator, research/2026-10-05-archive-org.md section B; arbiter, claim 3]. Both are known only through Oliver's text, and no reviewer weighed them. They show the brothers working together on line induction before the 1887 block. The tube result is what textbook electromagnetism predicts: equal and opposite currents on a closed coaxial return produce no field outside it (Ampère's law round the outside encloses zero net current) [memory, high]. The arbiter notes it may bear on the 1880 coaxial patent below, as its own inference; no source links them. - Arthur. Arthur later sided with Preece on loading. Heaviside in 1922, his own characterization 35 years after the event: "AWH was very afraid of P ... The greatest mistake AWH ever made was to throw me overboard" (IET SC MSS 068, per Bruton). - Stated reasons. The Electrician's editor gave space and libel reasons; those are better documented than pressure from Preece's circle, though none of the three is proved.

None of this removes the documented blocks. It shows a conflict with faults and stakes on more than one side.

Kept in the record, unweighed. British patent no. 1407 of 1880, "Improvements in Electrical Conductors", granted to Oliver alone (Arthur, as a Post Office employee, could not hold it), which Bruton calls "essentially the patent for ... a 'coaxial cable'" (recognised by C. S. Bright, Submarine Telegraphs, 1898); the specification is unread [web, via Investigator; arbiter, Investigator findings]. IET archive records seen only as catalogue entries, not read: Preece's "America Revisited - 1907" proof with Heaviside's manuscript notes (SC MSS 005/I/2/057), and the Stevenson correspondence 1897-1921 on "Heaviside's brother's career and being blocked by Preece" (SC MSS 005/I/6/43). Yavetz 2011, a later study, was seen as a search snippet only and is not reached [arbiter, Investigator findings].

On poverty, institutional relations and the outsider story [web, via Investigator; arbiter, Investigator findings]: Hunt 2012 reports that Heaviside refused a Royal Society Relief Fund grant in 1894 and received a civil-list pension (a state pension granted by the Crown) from 1896, GBP 120 a year, later GBP 220 (https://physicstoday.aip.org/features/oliver-heaviside-a-first-rate-oddity). The IET archives blog says he had been "expelled previously for not paying his membership fees" before the IEE made him an honorary member in 1908, and Bruton dates the striking-off to 1881 (https://ietarchivesblog.org/2023/04/05/from-under-the-sea-to-the-edge-of-space-the-work-of-oliver-heaviside/). A Google snippet of Mahon's 2009 biography (IET Digital Library) mentions a letter to Bethenod on why he refused the Hughes medal; the letter and its reason are unread (research/2026-10-05-google.md). Föppl's 1894 textbook carried his form of the equations (Hunt; see "Why 7a and 7c get different verdicts" in section 6).

Records that go quiet. In six places the record stops where you would expect it to continue. Following the review's rules, the arbiter treats each silence as something to explain, not as a neutral gap. The first Investigator run (6.5 minutes of web searching, 2026-10-05) has touched the first four; its findings are leads, not verdicts, and nothing was saved to the project [arbiter, Records that go quiet].

Genuinely open (from the arbiter). - How close is his 1893 gravity analogy (factors, signs) to linearised general relativity, and does his h field yield any result for a spinning body? (Answered in part: the "gravitational difficulty" of sec. 68 is that nobody knew how gravitational energy flows; App. B builds a moving-mass h field and a reversed energy flux, 4.2.) The deciding calculation: take page images of App. B (pp. 455-466; they also settle the printed exponent of u), compare his factors with linearised general relativity, work out the gyroscope frame-dragging rate his equations would predict for GP-B's orbit, and set it against the measured -37.2 ± 7.2 mas/yr (about 19% uncertainty). If his factors give a rate inside that band, Wikipedia's "confirmed by GP-B" would have real support; if outside, GP-B would count against his form [arbiter, What would settle it]. - Did his operational calculus give any wrong result in the divergent-series cases the Royal Society referee objected to, and did Bromwich have to correct any of his results? No source read here names one. Deciding records: the referee report on Part III (Royal Society RR series; Burnside named by a snippet only) and whether the modified Part III in Electromagnetic Theory Vol. 2 matches it. - Was Preece's 1887 block an engineering judgement that went wrong, a defence of his reputation and stake, or both; and was it before submission or a rejection by the Society? The block and his earlier public position are documented; the Heaviside article attributes it to protecting his reputation; the one primary glimpse (the catalogue description of his draft reply to Kelvin: "dreadful twaddle", Kelvin's formula) fits both a stake and a model that omitted inductance; the 1873 exchange and Kelvin's 1887 remark complicate it. Deciding records: the Preece draft SC MSS 022/VII/526 itself; the Society of Telegraph Engineers 1887 council minutes; IET notebook SC MSS 005/I/1/12; BT Archives (TCB 274/1, POST 30 files); The Electrician 1887 volumes (Biggs's removal); the 1887 manuscript. - What is Heaviside's printed formula in sec. 83, and does his split of permanent and induced magnetisation give a different internal stress that any measurement could reach? His argument is now read in transcription (4.2): the total force is not in dispute. His contents also list sec. 79, "Resultant Action on Solid Body independent of the Internal Stress, which is statically indeterminate" [read: contents p. xv]. Deciding check: page images of secs. 79-85 (pp. 103-107), then compare his force with F = ∇(m·B) and the Kelvin/Maxwell force densities inside a magnetised body. - What did Kennelly's March 1902 paper say, and did either man propose reflection rather than guidance? Nobody in this review has read Kennelly. Is the Wikipedia 1923 ionosphere date a sourced alternative or an error (Bruton and the IET blog give 1924)? Did Heaviside write on the layer earlier or later than 1902? - What is the true total of Pupin's payment ($455,000, over $400,000 or $500,000)? Deciding record: the AT&T/Pupin contracts. - Local energy flow: is there any measurement that could pick out S = E × H from S plus a curl? - Verify eqs. (47)-(48) symbolically once pp. 136-138 are obtained (a SymPy exercise you could do yourself). The full text of Electromagnetic Waves (1889) is now located on archive.org (electromagnetic01heavgoog), not yet opened. - What exactly did he argue against relativity, and where? Not in Electromagnetic Theory Vol. 3 by keyword; look in the Vol. 4 notes (2003 AMS Chelsea edition) and the 1903-1905 Nature letters. Only once that is read can it be matched against GPS timing, time-dilation and light-bending tests. Wikipedia quotes the historian Howard Eves that Heaviside "was the only first-rate physicist at the time to impugn Einstein, and his invectives against relativity theory often bordered on the absurd" [web] https://en.wikipedia.org/wiki/Oliver_Heaviside . That is Eves's characterization. Note that in the same sentence Eves ranks Heaviside as "first-rate".

---

8. Check yourself

1. In Heaviside's eq. (61), why does the H field jump at the sheet, but E (eq. 60) does not?
2. A thin film has sheet resistance R_s = 377 ohm per square (x = 0.5). What fractions of incident power are transmitted, reflected and absorbed?
3. Show that with R/L = G/C the propagation constant γ = √((R + jωL)(G + jωC)) becomes √(LC)(R/L + jω). What are α and the wave speed?
4. Heaviside's eq. (63) predicts a half-transmitting copper sheet 0.085 nm thick. Name two reasons this cannot be built, and say what could be built instead.
5. Using p = d/dt as an algebraic symbol, solve dy/dt + 2y = 6 with y(0) = 0. Check by substitution.

---

9. Further reading

Mainstream - D. J. Griffiths, Introduction to Electrodynamics: chapters 7 (Maxwell's equations), 8 (Poynting theorem), 9 (waves, reflection) [memory, high]. - D. M. Pozar, Microwave Engineering, chapter 2 (transmission lines, telegrapher's equations) [memory, high]. - J. D. Jackson, Classical Electrodynamics, units appendix (CGS vs SI) [memory, high]. - P. J. Nahin, Oliver Heaviside: Sage in Solitude (IEEE Press, 1988), the standard biography, including the Preece and loading-coil story [memory, high]. - B. J. Hunt, The Maxwellians (Cornell, 1991), how Heaviside, FitzGerald, Lodge and Hertz turned Maxwell's theory into modern form [memory, high]. - Wikipedia pages used in this review: Oliver Heaviside, Loading coil, Heaviside condition, Kennelly-Heaviside layer, Gravitoelectromagnetism, Gravity Probe B (URLs above) [web]. - B. J. Hunt, "Oliver Heaviside: A first-rate oddity", Physics Today (2012), https://physicstoday.aip.org/features/oliver-heaviside-a-first-rate-oddity : the 1887 block, the editor Biggs, the Operators rejection, Pupin's payment [web]. Also "Heaviside's Operator Calculus", Dead Reckonings, https://deadreckonings.com/2007/12/07/heavisides-operator-calculus/ [web]. - E. Bruton, "From theory to engineering practice: shared telecommunications knowledge between Oliver Heaviside and his brother and GPO engineer Arthur West Heaviside", Phil. Trans. R. Soc. A 376 (2018) 20170454, open access, https://royalsocietypublishing.org/rsta/article/376/2134/20170454/115714/From-theory-to-engineering-practice-shared : the brothers, Preece, the 1873 duplex quarrel, the 1880 coaxial patent, the floorboard papers; cites IET and BT archive records [web, via Investigator]. - IET Archives blog, "From under the sea to the edge of space: the work of Oliver Heaviside" (2023), https://ietarchivesblog.org/2023/04/05/from-under-the-sea-to-the-edge-of-space-the-work-of-oliver-heaviside/ ; IET archive catalogue (Heaviside and Preece papers), https://iet.adlibhosting.com [web, via Investigator]. - Grace's Guide, Oliver Heaviside (chronology and the 1925 IEE obituary), https://www.gracesguide.co.uk/Oliver_Heaviside [web, via Investigator]. - B. Mahon, Oliver Heaviside: Maverick Mastermind of Electricity (IET, 2009), cited by Hunt for the Biggs episode [memory, medium for the title].

Heaviside's own works (public domain [memory, high]; archive.org scans located by the Investigator [web, via Investigator]) - Electrical Papers, Vols. 1-2 (1892): early papers including the telegraph and induction work. Vol. 2 Sec. XL has the 1887 distortionless condition; Art. 38 is his 1887 reply to Preece. https://archive.org/details/electricalpapers01heavuoft ; https://archive.org/details/electricalpapers02heavrich - Electromagnetic Theory, Vol. 1 (1893): vector algebra, the circuital laws, the Poynting flux (sec. 70), the "gravitational difficulty" (sec. 68), magnetised matter (sec. 83, pp. 103-104), and the 1893 gravity article (App. B, pp. 455-466), https://archive.org/details/electromagnetict01heavuoft . Vol. 2 (1899): operational calculus, including his reply to the rigorists (sec. 425, p. 9), https://archive.org/details/electromagnetict02heavrich . Vol. 3 (1912): waves, moving dielectrics (sec. 473, Lorentz's equations), https://archive.org/details/electromagnetict03heavuoft . A 1971 Chelsea edition includes an account of his unpublished notes for a fourth volume [web] https://en.wikipedia.org/wiki/Oliver_Heaviside ; the 2003 AMS Chelsea edition with the Vol. 4 notes is on archive.org for lending only (not opened). - Electromagnetic Waves (1889), the full text containing pp. 139-143 and eqs. (47)-(48): https://archive.org/details/electromagnetic01heavgoog (located, not opened). - Encyclopaedia Britannica, 10th ed., vol. XXXIII (1902), "Telegraphy" (his ionosphere sentence; author initials garbled in the OCR), https://archive.org/details/in.ernet.dli.2015.209824 . - Letters to Nature, 1892-1905 (unread; where his relativity remarks may be). - For comparison on claim 6: A. E. Kennelly, "On the Elevation of the Electrically-Conducting Strata of the Earth's Atmosphere", Electrical World and Engineer 39(11): 473, 15 March 1902 (cited, unread). - "On Operators in Physical Mathematics", Parts I and II, sent to the Royal Society (1892-93) and printed (journal details not checked here): Part I https://gwern.net/doc/math/1892-heaviside.pdf ; Part II https://archive.org/details/philtrans01936980 [web]. - "On Electromagnetic Waves, and the Effects of ... Forced Vibrations of Electromagnetic Systems", Phil. Mag. (1888-89 [memory, medium]), the source of pp. 139, 142, 143.

---

Answers

1. The sheet carries a surface current K = σ d E (per unit width). Ampere's law around a thin loop straddling the sheet encloses that current, so tangential H jumps by K (by 4πK in Heaviside's units). Faraday's law around the same loop encloses only a vanishing area of changing B, so tangential E is continuous.
2. R_s = 1/(σ d) = 377 ohm means Z0 σ d = 1, so x = 0.5. E2/E1 = 1/1.5 = 2/3: T = 4/9 = 0.444. E3/E1 = -0.5/1.5 = -1/3: R = 1/9 = 0.111. A = 2x/(1+x)² = 1/2.25 = 0.444. Sum = 1.
3. R + jωL = L(R/L + jω) and G + jωC = C(G/C + jω) = C(R/L + jω). The product is LC(R/L + jω)², whose root is √(LC)(R/L + jω). So α = R√(C/L) = √(RG) (using G = RC/L) and β = ω√(LC), speed ω/β = 1/√(LC), both independent of frequency.
4. (a) It is about a third of one interatomic spacing, so it is not a continuous film; (b) bulk conductivity does not apply to films a few atoms thick (surface scattering raises resistivity). Instead build any thin resistive film with sheet resistance 188 ohm per square; the formula depends only on σ d.
5. (p + 2) y = 6, y = 6/(p + 2) · 1 = 3(1 - e^{-2t}). Check: dy/dt = 6e^{-2t}; dy/dt + 2y = 6e^{-2t} + 6 - 6e^{-2t} = 6. And y(0) = 0.