Mathematician review: Steinmetz (Charles Proteus Steinmetz), project "Steinmentz-Charles"

Role: PhD mathematician (Step 2). I check derivations, notation, units and hidden assumptions. Mainstream physical judgement is the Physicist's job; where a result depends on physics I say so and hand it over. Tags: [read] = source text or page image I read; [calc] = my own computation (Python/NumPy, scratch only); [web] = URL given; [memory] = recalled, confidence stated.

Source gap: none for text. The three books have full text layers. Only parts were deep-read (Discharges PDF pp.111-150; Alternating-Current PDF pp.179-215; Transient PDF pp.318-397, 486-545, 704-739). Everything else rests on digests. I added my own page-image reads: Alternating-Current PDF p.195 (corona formula) and Transient PDF pp.462 and 467 (radiation resistance). [read: page image]

Summary of verdicts

Corrections to Step 1 (and to the deep reads)

1. claims.md claim 2 and facts.json (Discharges entry) say Steinmetz gives no equation for the arc supply and does not use "negative resistance". True for Discharges (1914) pp.111-150, but incomplete for his whole work. The Investigator's finding is correct: in Theory and Calculation of Electric Circuits (1917) Ch. X secs. 96-99, printed pp.189-192, he writes the arc "has an effective negative resistance, -de/di ... and thereby generates alternating power, from the consumed direct-current power" [read: research/pages/ia-electric-circuits-1917.txt, around printed p.190]. I confirm it. He still writes in sec. 85 that the supply mechanism "is still practically unknown" [read: same file, p.166]. Fix: claim 2 should say the 1914 wording is the lag loop, and the 1917 treatment is quantitative and negative-resistance based.
2. claims.md claim 4: the f1 = 7.5e9/l0 law and its table are not first in the Transient book. They are in Steinmetz, Electrical World, 27 Aug 1898 (Investigator, [web: tuks.nl; research/files/steinmetz-1898-...pdf]). I did not re-read that scan.
3. claims.md claim 5 says the 24:1 ratio is "not reproduced" and flags it for me. I confirm and sharpen it (section on claim 5): the example is in a non-propagating (stop-band) regime and the printed alpha and beta look swapped relative to the book's own eq. (10); the "7.86 wavelengths" is off by a factor 4; and even with his own alpha, beta the sum is 19.4x, not 24x. The concept survives in a different regime.
4. Deep read (Transient, Section III ch. II, B6): it writes "E_open = E1/u". The book (PDF p.355-356, eq. 72-75 and the 283 example) gives E_open = (2/(pi u)) E1, which is what the 283 needs: 2/(pi x 0.00225) = 283 [read: p.356; calc]. The deep read's 283 is right; only its symbolic summary is wrong. This follows from E_open = E1/|cosh(gamma l)| with gamma l = alpha l + j pi/2, so |cosh| is about alpha l = u pi/2 [calc].
5. Deep read (Transient, C4): printed "300,000 V" at p.376 should be 150,000 V. I confirm independently: the series (sin t + sin 3t/3 + ...) equals pi/4, so e = (4/pi)(i0 z0)(pi/4) = i0 z0 = 750 x 200 = 150,000 V [calc]. Physical check: cutting a uniform current i0 off at the open end launches a step i0 z0, with no doubling, since the open end already has zero current [calc, standard line theory].
6. Deep read (Alternating-Current): the corona formula P = a(f + c)(e - e0)^2 with A = 241 was flagged as missing a scale factor. Confirmed on the page image (PDF p.195, printed p.166): A = 241 with no 10^-5. Peek's formula carries 241 x 10^-5 [web: https://eepower.com/technical-articles/understanding-corona-discharge-in-high-voltage-transmission-lines/ and search results listing "241 x 10^-5 x (f+25)/delta x sqrt(r/D) x (V-V0)^2"]. With 10^-5 the deep read's example gives about 19 kW/km, a sane figure; without it, megawatts. Treat the printed A as a misprint or lost multiplier in this edition. I did not check other editions.
7. Deep read (Alternating-Current, 1.4): the open items (maximum power factor and the OCR "g = 2/2 xfC", "po = 4") are resolved by my calculation. For two layers of equal capacity C, only layer 2 leaking with conductance g, the power factor peaks at g = sqrt(2) omega C = 2 sqrt(2) pi f C (the deep read's reconstruction is right) with maximum power factor 1/3 [calc]. I did not see the printed value on the image, so "1/3" is mine, not the book's.
8. facts.json/claims.md claim 7 "about 0.2 ohm per metre of No. 00 wire at 100,000 cycles": confirmed from the page image (Transient PDF p.462, eq. 10: r2 = 2 pi^2 l f 10^-9 = 1.97 l f 10^-8 ohm with l in cm; p.467 gives the same per metre as 1.97 f 10^-6). At f = 1e5 and l = 100 cm: 0.197 ohm [calc]. Note that the formula is for a section of an infinitely long conductor without return conductor, and Steinmetz himself says (p.467) that below about 10^4 cycles "the condition of a conductor without return conductor can hardly be realized".
9. Minor: the deep read's Fig. 94 numbers. In Electric Circuits 1917, the arc characteristic of Fig. 94 is e = a + b/sqrt(i) (equation constants lost in OCR). With R(0) = 18.5 ohm at 3 A (so b = 192) the formula gives 19.9 and 26.6 ohm for swings of 1 and 2 A, against printed 20.3 and 27.5 (2% and 3.5% off) [calc]. Likely graph-reading or a slightly different constant; not a substantive error.
10. Notation slip in the Transient book (deep read agrees): p = s + u (eq. 27) against exponents u - s, u + s afterwards; "h^2 > LC s^2" (p.520) should read h^2 < LC s^2; "R1" stands for R1^2. These do not affect any result [read; deep read's sympy check].

Claim 1. A sustained or growing oscillation needs a continuous supply, which comes from the power source through an arc

a. Consistency. Yes. Write W for the oscillation's stored energy. Then dW/dt = Ps - Pd. A transient has Ps = 0, so W decays; continual means Ps = Pd; cumulative means Ps > Pd until Pd(W) grows to meet it. This is the energy statement of Discharges p.142 (Steinmetz prints no formula; the formula is the deep read's paraphrase) [read: p.142]. The remark "frequency transformation, from direct-current or low-frequency energy to the oscillation frequency" is mathematically necessary: a linear time-invariant element cannot move power between frequencies, so the supplier must be nonlinear or time-varying. An arc is both [calc/standard]. b. Experiments. Arc oscillators (Duddell 1900 singing arc, Poulsen) [memory: high]; arcing grounds recorded in oscillograms (his Figs. 44-46, 59-65, from 22 kV, 82-107 kV systems and Cunningham's artificial line, Proc. AIEE 1911) [read: deep read p.120, 143-145]; contemporary operating experience of arcing grounds in the J. AIEE of May-June 1923 and the Commonwealth Edison 1919 oscillation investigation (Investigator [web: WorldRadioHistory J. AIEE 1923; HathiTrust]). The Investigator's sources document the phenomenon, not Steinmetz's mechanism. c. Tested. Yes in the sense of observation and engineering practice; replicated by others (Duddell, Poulsen, wireless engineers) [memory: high; not re-sourced because no verdict here turns on it]. d. If false: mainstream would need to give up energy conservation. Nothing changes if true. Verdict: Established.

Claim 2. The arc supplies the energy through a lag, a "negative energy cycle or reversed hysteresis loop"

What is in the sources. 1914 Discharges pp.146-148: overlapping oscillation trains need "a phase displacement or lag within the oscillation, which gives a negative energy cycle, or reversed hysteresis loop"; Fig. 66 (Ti-C arc, 0 to 4.1 A and back in 1/120 s) shows higher voltage on rising current. He says the mechanism is "still practically unknown" for circuits [read: p.146-147]. 1917 Electric Circuits Ch. X: negative resistance from the falling characteristic, with the lag limiting amplitude; 1923 Trans. AIEE 42:470, abstract: "a negative resistance in combination with a source of power" [web: https://ieeexplore.ieee.org/document/5060887; Investigator; full text not read by anyone].

a. Maths. The 1914 statement fails three tests. (i) Units. The area of the loop in the (i, v) plane is the integral of v di, in watts, not joules. Energy per cycle is the integral of v i dt, which is not an area in that plane. The magnetic analogy (the integral of H dB, energy density) only works when the axes are chosen as conjugates in the energy sense, and Fig. 66 does not do that [calc; deep read made the same remark]. (ii) Decomposition. Take a small current oscillation i = cos t and arc voltage v = -R i/(1 + j tau) (a falling characteristic with static slope -R and a first-order lag tau). Then per cycle: the energy given to the circuit is pi R/(1 + tau^2) (in-phase part); the loop area is pi R tau/(1 + tau^2) (quadrature part). Numerical check: tau = 0: energy 58.1, area 0; tau = 1: energy 29.1, area 29.1; tau = 5: energy 2.2, area 11.2 (R = 18.5) [calc]. So the loop area is the reactive (quadrature) part and the supply is the in-phase part; lag lowers the supply by 1/(1 + tau^2). Lag alone cannot create supply: it needs the negative in-phase slope, i.e. the falling characteristic. (iii) Direction. In that same model, voltage is higher on rising current (the clockwise loop of Fig. 66) [calc]. So Fig. 66 is exactly what a falling characteristic plus thermal lag gives. It does not require, and does not show, that the loop itself is the source. The 1917 account (negative dynamic resistance -de/di, lag limits amplitude) is consistent with this calculation. Hence Steinmetz's 1914 wording is incomplete rather than wrong in its data; his later account fixes it. b. Experiments. Static arc falling characteristics are textbook (Ayrton, Duddell) [memory: high]. Fig. 66 is one measured loop; Steinmetz gives no table. His notebooks reportedly contain "Frequency transformation by third class conductor" calculations (March 1922) and "tests on titanium and carbind" (1912) [web: Schenectady finding aid, research/files; not examined]. c. Tested. The falling-characteristic picture is tested and replicated in arc-oscillator practice; no neutral test of the loop-as-supply wording exists. d. If true as worded: nothing in mainstream would change; the corrected form is mainstream. Verdict (split): arc supplies power via its falling characteristic = Established; "the lag/loop is the supply" = Refuted by calculation (as a sufficient mechanism; lag lowers supply by 1/(1+tau^2)). Retained for the record: Steinmetz openly flags the mechanism as unknown (1914, 1917) and the "arc has no frequency of its own" statement (p.149) is correct for a negative-resistance oscillator whose frequency is set by the tank [read; standard].

Claim 3. General circuit equations, constants u, m, s, surge impedance, distortionless condition

a. Maths. I checked the deep read's account against its sympy checks and re-derived key steps by hand. The telegrapher's equation d2i/dl2 = rg i + (rC + gL) di/dt + LC d2i/dt2 follows from de/dl = r i + L di/dt and di/dl = g e + C de/dt [read: p.489]. The trial solution gives a^2 = (bL - r)(bC - g) and z = (bL - r)/a [read: p.490]. With u = (r/L + g/C)/2, m = (r/L - g/C)/2, b = u -/+ s: (u - s)L - r = -(m + s)L and (u - s)C - g = -(s - m)C, so z^2 = (L/C)(s + m)/(s - m) [calc]. This is the book's c = sqrt((s + m)/(s - m)). The distortionless condition m = 0 (r/L = g/C) is Heaviside's (1880s) [memory: high]. In the high-frequency limit h = s sqrt(LC), k = q sqrt(LC): the front travels at 1/sqrt(LC) and decays as e^(-ut) whatever the frequency [read: p.539-540]. Interpretation. The "energy transfer constant" s is mathematically the spatial exponential gradient of the pulse amplitude (h = s sqrt(LC)) expressed as a rate. Saying one component "drops energy and the other picks it up" is a bookkeeping reading of an exponentially graded wave front: at a fixed point the energy decays at 2(u - s), while the pulse as a whole decays at 2u. It is consistent but adds no new equation. Hidden assumptions: r, L, g, C constant and frequency-independent (his single stated assumption, p.488). He himself treats the breakdown at high frequency (skin effect, Section V, [read: contents/digest]). A frequency-dependent r or g is no longer "constant", which strains the assumption for impulses (deep read). Radiation has no separate term; it enters only through effective r and g. Two notational notes: sign convention de/dl without a minus sign (equivalent to reversing l); partial derivatives written as ordinary d. The restriction s^2 <= u^2 (Ch. II) with the non-periodic condition s^2 > m^2 leaves a window that closes when g = 0 (u = m). Ch. III does not repeat the restriction (deep read; I agree). b. Experiments. Transmission-line theory is among the best-confirmed in engineering (telegraph and telephone lines, matched terminations); Steinmetz's own oscillograms of reflection on Cunningham's 130-mile artificial line (Figs. 51-53) are consistent with it [read: deep read p.128-130]. His patent US 1,230,615 (filed 1913, granted 1917, GE) uses a shunt resistor equal to the surge resistance sqrt(L/C) so waves pass out of a transformer without reflection; matched termination gives zero reflection coefficient, which is standard [web: https://patents.google.com/patent/US1230615A/en; Investigator; I did not re-read it]. c. Tested. The equations: yes, widely, by others. The interpretive names for s: no test (no measurable difference). d. Nothing in mainstream changes; it is mainstream circuit theory with different vocabulary (Steinmetz's "attenuation, distortion, energy-transfer" constants = Heaviside's attenuation/distortion plus a pulse-shape constant). Verdict: Established (equations); "energy transfer" reading = Plausible, untested (interpretation, no extra observable).

Claim 4. Line resonance and surge effects

a. Maths. Quarter-wave line: f = S/(4l) with S = 188,000 miles/s gives 47,000/l cycles; 1880 miles at 25 Hz and 783 miles at 60 Hz [calc: 188,000/(4x25) = 1880; /(4x60) = 783]. With r = g = 0, e and i of a quarter-wave line obey E1 = -j sqrt(x/b) I0, so constant voltage in gives constant current out (standard) [read: p.351]. Open-circuit magnification: 2/(pi u) with u = r/(2x) = 0.00225 gives 283 [calc; book p.356]; 283 x 5000 V = 1.415 MV [read]. This is a lossy-line result for a single-frequency drive; real coil constants vary with frequency, and the example uses shunt capacitance only (the book moves to series capacity in the next chapters). It is an internally consistent model, not a measurement. f1 = 7.5e9/l0: c/(4 l0) with c = 3e10 cm/s [calc]; the table (4700 at 10 miles to 470 at 100 miles) reproduces [calc]. The statement "independent of wire size and height" follows because L0 C0 = l0^2/c^2 for a lossless line over ground; it ignores ground and wire losses and corona. Short-circuit interruption: should be 150,000 V, not 300,000 V (correction 5). The conclusion (surge of order i0 z0 above line voltage) stands. Unmeasured claim: "danger frequencies" 10,000-30,000 (transformer) and 30,000-100,000 cycles (coils) [read: p.390]; no measurement cited there. He repeats them in 1917 as "frequently observed" without citing data (Investigator). The 60,000 kW / 700-mile design is a design calculation; constants reproduce: sqrt(L/C) = 185 ohm, u = 0.066, 306 A at 69 kV [calc by deep read; checked arithmetic]. It was not built, as far as the pages say. b. Experiments. Line resonance, travelling-wave reflection and the unsuitability of long lines for voltage regulation are textbook [memory: high]. Steinmetz's 1898 note already used the table (Investigator). c. Tested. Quarter-wave behaviour: yes, widely (it underlies Tesla-coil style resonant transformers and line-length effects) [memory: high]. The transformer-winding danger frequencies: no measurement in the pages read. Whether the 1917 text cites measurements is unchecked. d. Nothing in mainstream changes. Verdict (split): resonance formulas, f1 law, 283x example as a model = Established; 10-100 kHz danger frequencies as cited = Never properly tested (not dismissed; simply no data given here).

Claim 5. Lightning flashes in clouds as discharges in a series-capacity chain (24:1)

a. Maths. I recomputed section 47 (Transient PDF p.397, the page image was read by the deep reader) with Steinmetz's constants: L = 2e-8 H, C0 = 1e-13 F, C = 4e-11 F, r = 1 ohm, g = 4e-6 mho, f = 1e8 Hz, n = 300, e0 = 30 kV. [calc] - x = 12.57 ohm, xc = 39.79 ohm (printed 12.6, 39.7), b = 62.8e-6 mho. Z = 1 - 27.2j ohm, Y = (4 + 62.8j)e-6 mho: matches the printed values. - Z Y = 1.714e-3 - 4.6e-5 j: nearly real and positive, because the series arm is net capacitive (x < xc; the chain's series resonance is at 178 MHz). sqrt(ZY) = 0.0414 - 0.00056j. That is an exponentially decaying, essentially non-propagating field (a stop band). Evaluating the book's own eq. (10) as printed also gives alpha = 0.0414, beta = 0.00056 [calc]. The printed alpha = 0.0021, beta = 0.0412 are those two numbers exchanged (the deep read found the same). - With the printed alpha, beta, voltage across the 300 gaps varies from about 200 V to 3050 V and the sum of the magnitudes is 19.4 e0 (582 kV), not 24 e0 (730 kV). Steinmetz got 730 kV by multiplying 300 by the mean of his two end values (2280 and 2560); he did not integrate. His end values (64.6 A, about 2570 V) match my evaluation with the printed alpha, beta at one end of the chain only [calc]; I cannot say which orientation he used. The book's closed form 35,500 sqrt(e^-0.0042 l + 0.08 e^+0.0042 l - 0.568 cos(24.72 - 0.0824 l)) gives 26.6 kV at l = 0 against the stated 30 kV [calc]. - "7.86 complete wave-lengths": with beta = 0.0412 the 300 cylinders contain 300 beta/(2 pi) = 1.97 wavelengths. 7.86 = 24.72/pi, four times too many (deep read; I confirm) [calc]. - With the correct alpha, beta the infinite-chain gap voltage is e0 xc sqrt(y/z) = 1815 V falling to nearly zero within a few tens of cylinders; the sum is about 44.7 kV, ratio 1.5 [calc]. - The concept. Sum of the capacitor (gap) voltages can exceed the terminal voltage only because the series inductor drops are of opposite sign. This needs a net-inductive or near-resonant series arm. At 178 MHz (series resonance) my model gives a sum of 28.8 e0, and at 250 MHz 10.9 e0 [calc]. So the principle is mathematically available but his example puts the numbers in the wrong regime. And "the sum of gap voltages" is an arithmetic sum of magnitudes at different positions, not a potential difference. The terminal potential is still e0; what matters for breakdown is the local gap voltage, so the quantitative "few hundred million volts" extrapolation (24:1 times what?) is not computed from any cloud parameters. He gives none. Hidden assumptions: a regular chain of identical elements; lumped inductance and resistance in the drop path (not shown to exist in a cloud); steady sinusoidal drive at 100 MHz applied to a discharge process; resistive and corona losses small. The cloud model is stated as "probable" (p.393). b. Experiments. None for clouds in the source. The Investigator found Steinmetz's 1922 laboratory lightning generator (10,000 A, over 100,000 V, a hundred-thousandth of a second, AP report, NY Herald, 3 Mar 1922 [web: https://www.loc.gov/resource/sn83045774/1922-03-03/ed-1/?sp=6]); its numbers are self-consistent (about 10^9 W, about 10 kJ: Investigator's arithmetic) but it does not test the cloud-chain hypothesis. Mainstream cloud-electricity data (leaders, charge layers) belong to the Physicist. c. Tested. Not tested for clouds. The multigap arrester's frequency-dependent division of voltage is engineering practice (A.I.E.E. Trans. 1906-07, as cited) [read: p.393]. d. If true: the picture of a cloud as a chain of drop-to-drop discharges with an inductive path would be required; mainstream stepped-leader models would need a displacement-current/ladder component. That is for the Physicist. Verdict (split): printed worked example = Internally inconsistent (alpha and beta exchanged, wavelength count wrong by 4, sum of 24 not reproduced); cloud-chain hypothesis = Never properly tested.

Claim 6. Dielectric loss and corona

a. Maths. Constant power factor implies loss proportional to f e^2: with i proportional to f e, P = pf e i proportional to f e^2 = n f D^2 [read: p.180; calc]. The D^1.6 law is for rotating fields, a separate small effect (deep read's correction 3 stands). Leakage in a homogeneous dielectric: pf about gamma/(2 pi f k), inversely proportional to f [read: p.181-182; standard]. Two layers: the Maxwell-Wagner effect. For equal capacities and one leaking layer, I find pf maximal at g = sqrt(2) omega C, maximum 1/3, and within 10% of that maximum only over about a factor of 2.7 in frequency (0.43 to 1.16 times g/omega) [calc]. So it gives "approximately constant" pf only over a narrow range; a wide-range constant pf (decades) needs a distribution of time constants (many layers), which the book does not discuss. Steinmetz says "over a considerable range". Glass plates: the book's formula gives 11.4 kV/cm (glass) and 45.5 kV/cm (air) at 25 kV, not the printed 8.4 and 35.7 [calc: 25/(0.6/4 + 0.4)]. Check: 0.6 x 8.4 + 0.4 x 35.7 = 19.3 kV, not 25. Conclusion (air breaks down) unchanged. Breakdown 30 kV/cm peak = 21.2 kV rms [calc: 30/sqrt 2]. The effective-conductor critical voltage e0 = m0 g0 delta r ln(s/r) is the surface field of a thin cylinder reaching g0 [calc: standard electrostatics]. Corona formula: printed A = 241 lacks the 10^-5 (correction 6); with it, a 100 kV line (r 0.5 cm, spacing 200 cm, m0 = 0.9) loses about 19 kW/km [calc, deep read]. "Energy distance" (corona starts above the surface-breakdown field because a finite volume must be over-stressed) is a qualitative hypothesis, stated without data in these pages [read: p.194]. b. Experiments. Peek's measurements underlie the corona formula (cited: Peek, "Dielectric Phenomena in High-voltage Engineering", p.200) [read]; the formula is still taught [web: eepower.com link above]. Constant-power-factor losses in solid insulation are widely observed (Curie-von Schweidler / "universal dielectric response") [memory: medium; not web-checked since no verdict turns on the attribution]. Corona starts above the Peek critical voltage: Peek's formula matches that experience. c. Tested. Peek's formula: tested by Peek and used by engineers since [web]. Steinmetz's two-layer explanation of constant pf: not tested in these pages. d. Nothing in mainstream changes. Verdict (split): scalings and corona formula (with the 1e-5) = Established; layered dielectric imitating constant power factor = Plausible, untested (narrow range as computed above).

Claim 7. Radiation as effective resistance through finite field velocity

a. Maths. Eq. (10), p.425: r2 = 2 pi^2 l f 10^-9 = 1.97 l f 10^-8 ohm (l in cm) for a section of an infinitely long conductor "without return conductor"; with return at distance D', r3 = 263 f^2 D' l 10^-18 ohm (cases (a) D' = 182 cm and (b) D' = 1820 cm give 0.048 and 0.48 f^2 x 10^-12 ohm per cm) [read: page images PDF p.462, 467]. Arithmetic: 0.197 ohm per metre at 100 kHz [calc]. Dimensional and arithmetic consistency: yes (re-expressed, 1.97e-8 l f equals 592 (l/lambda) ohm with c = 3e10 cm/s [calc]). Scaling check, [memory: high] standard antenna theory: a short current element has radiation resistance 80 pi^2 (l/lambda)^2, proportional to f^2. Steinmetz's linear-in-f result for "no return" is for a long conductor segment; with lambda = 3 km and l = 1 m the short-element value is 8.8e-5 ohm against his 0.197 ohm. The two are different models (his segment belongs to an infinite wire with a field that continues beyond the segment). He ends his discussion by saying the no-return case cannot be realised at low frequencies (p.467). I do not decide which model is physical; that is the Physicist's task. The hidden assumptions (uniform current along an infinite wire, no return, field of a round wire of equal circumference, velocity functions sil and col from Glaisher's tables) are the points to check. I did not re-derive the sil/col integrals; the appendix has seven misprints and table errors per the deep read. b. Experiments. No measurement is cited in the pages I read. Radiation resistance of antennas is established in general [memory: high]. c. Tested. Not in this book. The Investigator notes Dollard's later use of Steinmetz's finite-velocity-radiation-as-energy-component idea [web: OSE wiki copy of Dollard 1990; Dollard and his publishers are the interested party; mainstream engineering does not accept Dollard's wider claims, per memory: high]. Recorded as that characterization, not as a reason to exclude the work. d. If the magnitude were right for ordinary lines: line losses at 100 kHz would be dominated by radiation, which mainstream antenna theory would have to confirm; the Physicist should judge. Verdict (split): existence of a radiation resistance = Established; Steinmetz's magnitude for a wire "without return" = Never properly tested.

Investigator's findings, kept in the record

• Inventor's own work that bears on claims 1-2: Trans. AIEE 42 (1923) 470-477, "Frequency conversion by third class conductor and mechanism of the arcing ground and other cumulative surges" [web: https://ieeexplore.ieee.org/document/5060887, abstract only]; Electric Circuits 1917 Ch. X secs. 85-101 [web: https://archive.org/details/theorycalculatio00stei; page text read by me]; "Instability of Electric Circuits", Proc. AIEE 33(1) Jan. 1914 [web: https://ieeexplore.ieee.org/document/6661036]; the 1898 Electrical World table (claim 4); the 1912 GE Review address (cumulative oscillations "the most dangerous"; his ether as gas and solid is, by the Investigator's note, a contradiction) [web: tuks.nl]; US 1,230,615; Commonwealth Edison trouble investigation (1920); "Outline of Theory of Impulse Currents" (1916), "Lightning Phenomena in Electric Circuits" (1907), "Overdamped Condenser Oscillations" (1924) not opened.
• Supporting information by others: 1923 J. AIEE discussion of arcing-ground suppressors (Creighton) and neutral grounding; Slepian and Peters (Westinghouse, 1923, a competitor); Gilkeson 1933 quoting the 1923 abstract. These support that the problem was real and the negative-resistance account was taken up; not read in full.
• Related work: Dollard (1982, 1990, 2013) builds on Steinmetz's field symmetry and radiation components; Duddell and Poulsen arcs are the standard negative-resistance route.
• Historical record: Schenectady County Historical Society finding aid lists notebooks "Frequency transformation by third class conductor" (23 Mar 1922), "Nernst glower as oscillator", "tests on titanium and carbind" (1912), air-breakdown experiments (1922). Not examined; they could show whether Steinmetz measured the arc's in-phase negative resistance himself. No sign of the record going dark (Steinmetz published on this to his death in Oct 1923). No "absence of evidence" case here.

Open for later steps: full text of the 1923 and 1914 papers (does the 1923 paper contain the in-phase/quadrature analysis above?); Steinmetz's 1917 sec. 96-99 equations for the arc (the OCR dropped constants); the 1917 and 1898 sources for the transformer danger frequencies; any later check of the corona formula in other editions.