RankingsPrimerArbiterPhysicistMathematicianContrarianQESites logClaimsSourcesResearch notes

Peek-Frank: Mathematician's review (Step 2)

Reviewer: PhD mathematician role. Scope: derivations, notation, units, hidden assumptions. Physical and experimental questions are the Physicist's and Contrarian's; I answer a-d from the mathematics outward and say when I leave my lane.

Tags: [read: p.N] = PDF page of Peek, Dielectric Phenomena in High Voltage Engineering, 2nd ed., 1920 (PDF = printed page + 23 in Ch. III, +39 in the appendix). [read: page image] = the Step 1 deep readers' page-image reads, which I re-used where I did not re-render. [calc] = my own computation (scripts in the session scratchpad: m.py). [web] = URL given. [memory] = recalled, with confidence.

Source gap, repeated from claims.md: the text is OCR on 324 of 325 pages. I read the deep-read file in full, the claims and facts, the Investigator's notes, and I went to the extracted text of Chapter V (PDF pp.155-171, Line A / Line B and the c^2 fit) myself, because Step 1 had it on digest only. Nothing here rests on [memory] alone. The Investigator saved no files in research/files/ (the folder does not exist); its findings are leads from abstracts and snippets only.

Corrections to Step 1

  1. Line A is single-phase (settles an open item in claims.md). Step 1 left open whether Line A was single- or three-phase. The Chapter V figures label the same voltage axis as "Kilovolts between lines 110 ... 210" over "To neutral 55 ... 105", i.e. line voltage = 2 x neutral voltage, which is single-phase [read: p.159-160, text layer of Fig. 115-116]. Line A is four 3/0 cables (1.18 cm diameter, 7-strand), "Total conductor length 109,500 cm", spacing 310 cm, and Peek's fitted curve for it is p = 0.0117 (e - 72.2)^2 [read: p.159]. So the fitted disruptive voltage is e_o = 72.2 kV to neutral = 144.4 kV between lines, which matches the "about 145-150" Step 1 read by eye. My formula value is 70.0 kV with m_o = 0.87 and delta = 1.032 [calc]: 3% below the fit, which is inside the book's own scatter (below).
  2. The A = 241 versus 244 "misprint" is not a misprint in any meaningful sense. Step 1 and the deep read treat it as an unexplained 1.2% inconsistency. The constant comes from the fit of c^2 = A (r/s)^(1/2)-type relation in Table LV (12 tests, 4.5 to 8.8 in ln(s/r)). Refitting those 12 printed points myself, ln c^2 against ln(s/r) has slope -0.498 +/- 0.021 and residual rms 7.4% in c^2 (max 16%) [calc; data [read: p.171]; the OCR header says "x 10^8" but the numbers fit a 10^5 scale, which I infer from reproducing A]. Fixing the exponent at -1/2 gives A = 243 [calc]. So 241 and 244 are both inside the fit's own 7% scatter; the difference is rounding at two stages of the book, not an error of substance.
  3. Claim 1 lists "independent of ... humidity" among things the constants are independent of. The book asserts it twice (p.68, p.93) but pp.60-92 contain no humidity data [read: deep read section 6]. The only humidity statement with a number in the pages I read is Table I of the appendix (needles, humidity 34 percent, one value). Treat humidity-independence as asserted, not shown.
  4. Claim 1 states "g_o about 30" and the fit as 1 + 0.301/sqrt(r)." The printed fit is 29.8(1 + 0.301/sqrt r) for parallel wires, 31(1 + 0.308/sqrt r) for cylinders, and the working formula uses 30. Table VIII's "calculated" column mixes 31 with 0.301 rather than 0.308 [read: deep read]. These are rounding-level, but "about 30" hides a 4% spread between the three geometries (29.8, 31, 27.2 for spheres), which Peek attributes to field "balance" without calculation [read: p.73, 79, 88].
  5. Claim 3 says "A worked 100-mile line (e_o = 52.0 kV) recomputes correctly." Confirmed (51.90 to 52.01 kV, 0.2%) [calc, from deep read]. But claim 3's "Corona ... economic rather than physical limit" is Peek's forecast, not a calculation, and the Leadville and Au Sable comparisons cannot be recomputed from the book's pages (no complete inputs for Au Sable; Leadville points are plotted only). I would add "not recomputable" to claim 3.
  6. Claim 5: "55 to 900 kc" is not a frequency of a continuous wave. Each is a single half-sine pulse (a few microseconds) [read: p.302]. Step 1 says so correctly; the labels are also not exact: from the printed L and C the undamped LC frequencies are 45 kHz (labelled 55), 285 kHz (labelled 350), 781 kHz (labelled 900), 101 kHz (labelled 100) [calc]. Labels are round numbers within about 20%.
  7. facts.json "the deep read found 10 of 19 points were kept" and claims.md "9 of 19 points discarded": consistent and confirmed by my reconstruction (see claim 1). The book does not print which points were dropped; the list is inferred.
  8. The Table I "Average, 99.0" is actually 98.4 [calc]. Immaterial but it is a printed arithmetic slip.
  9. Nothing in facts.json is contradicted otherwise. The statement that "bundling decreases the critical voltage" (p.246) as a probable misprint is reasonable; the book elsewhere reports the opposite for split conductors (16-30 percent higher, digest p.95-98).

What the mathematics of the book actually is (short orientation for the reader)

Everything rests on one geometric fact. For two parallel wires of radius r with centre spacing s (s >> r), the electric field at distance x from a wire's axis is about E(x) = e / (x ln(s/r)), where e is the voltage to neutral. At the wire surface (x = r) this is g = e / (r ln(s/r)). Peek's "critical gradient" is the value of g at which the wire starts to glow. This is a worked-out formula from first-year electrostatics (Griffiths ch. 2); I recomputed Table I from it and it reproduces every printed g_v to within 1.1 kV/cm [read: deep read; calc].

Worked number. Wire radius r = 0.1 cm: Peek's law gives g_v = 29.8(1 + 0.301/sqrt 0.1) = 58.2 kV/cm, about twice the 30 kV/cm "bulk" strength. A thin wire must be stressed harder before it glows.

Claim 1. Peek's law of visual corona: g_v = g_o delta (1 + 0.301/sqrt(delta r))

a. Internal consistency. - The algebra is sound and the notation hazards are minor. In eq. (24) the letter a is the coefficient 0.301 and two lines later a is the distance 0.301 sqrt r; the coefficient 0.301 must carry units cm^(1/2) (unstated). Eq. (20) puts delta in both g_v and a prefactor as printed, and the intended meaning is clear [read: page image p.68, 72]. - The Sigma-Delta (group-average) arithmetic is correct. From Table IV: sum differences 158.5 and 17.63 give slope 8.99, intercept 29.84, ratio 0.3013 [calc]. The method is the standard two-group (Wald/Bartlett-type) line fit; it is legitimate and unbiased in the usual way, but it uses only the group means. Peek's claim that it is "as reliable as, or more so, than ... least squares" is not true as a general statement (least squares uses all the information; Wald estimators have larger variance), though for data this clean the difference is small. - The discarding is the weak point. 9 of 19 points were dropped "as probably in experimental error"; the criterion is not stated and the kept and dropped points have overlapping residuals from the final line (kept -5.5% to +4.0%, dropped -6.2% to +4.3%) [read: deep read]. I checked what the discards cost. All 19 points, least squares: g_o = 30.4, ratio 0.286 (against 29.8 and 0.301) [calc]; sorting by 1/sqrt(r) and group-averaging all points: ratio 0.302-0.308 [calc]. Rms misfit of Peek's law on all 19 points is 3.6%, of the all-point least-squares line 3.6% [calc]. So the selective discarding changed the constants by a few percent only; it did not manufacture the law. The honest reading: the constants are fitted to about +/-3%, and no reader should quote 0.301 to three figures. - Is the exponent 1/2 supported by the data? I fitted g_v = g_o(1 + c/r^n) with n free to the 19 printed points: best n about 0.41 (bootstrap 5-95%: 0.40-0.45), with g_o driven down to about 22 and rms misfit 2.8% rather than 3.6% [calc; crude optimiser, indicative only]. That is, with g_o left free the data prefer a slightly weaker power than 1/2, but then g_o no longer equals the 30 kV/cm plane value that Peek's reasoning needs. Fixing n = 1 is clearly rejected (rms 11%); n = 0.7 is rejected (6.4%). The 1/2 power is a good, not unique, description. Peek's statement that he plotted trial functions until a straight line appeared in 1/sqrt(r) [read: p.74] is therefore fair, with the caveat that other exponents near 0.4-0.5 fit at least as well. - In-sample testing: the "generally less than 1 percent" fit quality (Tables VI-VII) is true but those data were part of the fit [read: deep read; calc].

b. Consistency with experiments. Within my lane (data in the book and the Investigator's lead): the density factor holds to about 1-2% from delta 0.15 to 1.0 in Table VIII, with the two lowest-density points (delta 0.069, 0.141) 8% below the formula, without comment [read: p.80; calc]. Whitehead's 1917 compilation (as printed in the book's Table IV) gives a.c. critical gradients within 1.5% of Peek's [read: page image p.296]; this is another laboratory's data, quoted by Peek, an interested author. Fitch (Johns Hopkins, 1913) is reported by the Investigator to find that Peek's original equation "did not closely match", that he modified it, and that his results "support Peek's empirical relationship" [web: https://archive.org/details/influenceofdensi00fitcrich, via fetch summary, not checked by eye]. Both statements are kept here; the modified formula needs a page-image read. Whitehead's data show d.c. positive-wire corona about 4-6% below the a.c. peak, while Peek's own d.c. tables show equal polarity values [read: p.296]; the book does not reconcile this.

c. Tested? By Peek and GE helpers on polished wires indoors (1910 onward), by Watson (d.c.), Whitehead and Brown, Farwell, Fitch, all cited in the book; the 1984 Hartmann paper later compared a streamer-criterion calculation with "a variety of experimental results" [web: https://ieeexplore.ieee.org/document/4504655, abstract as recorded by the Investigator; my own fetch of the page returned no content]. The Protsenko-Trotsenko 2014 centenary paper shows the formula is still used [web: https://kpi.ua/nv-abstract-14-6, snippet only]. I cannot say from the record that a neutral replication of the full g_o and humidity independence exists; the book's own tests are the main source.

d. If it were true: nothing in mainstream physics changes; it is an engineering fit that mainstream breakdown theory explains (see claim 2).

Verdict: split. (i) The empirical dependence of onset gradient on wire radius and air density: Established as an engineering fit to about +/-3-4%, with the constants and exponent known only to that precision. (ii) "Independent of humidity and material": Never properly tested in the pages (material: five pairs at matching diameters in Table II is a coarse test; humidity: no data printed), not dismissed. (iii) The strict "constants independent of geometry": contradicted by the book's own 29.8 / 31 / 27.2.

Claim 2. The energy-storage distance (a = 0.301 sqrt(r) cm)

a. Internal consistency. This is where the mathematics is most instructive. - The "constant gradient at a distance" is an identity, not a derivation. From eqs. (21)-(22), g_v/g_o = (r + a)/r, so a = r(g_v/g_o - 1). For any measured curve g_v(r) and any chosen g_o, a distance a(r) exists. So "the experimental values bear out the above theory" (p.72) cannot fail. The testable content is only that one g_o and one simple form for a(r) fit all radii, spacings and densities [read: p.72; deep read; confirmed by my own algebra]. - The field formula used, E = e/(x ln(s/r)), is the field of an isolated line charge; it neglects the second wire's contribution and is good to O(r/s). Peek notes for spheres that the energy zone is distorted at small spacings and his own Fig. 48 shows a = 0.27 sqrt(R) does not give a constant gradient across spacings; he says g_o and the coefficient "must also be a function of X" [read: p.90]. That is an honest statement of a limit. - The sphere "energy distance" 0.27 sqrt(R) comes from the fitted 0.54 via approximation (30) which Peek says holds only at a = 0 [read: page image p.88-90]. The cross-check between the independently measured onset spacing X = 0.54 sqrt(R) (eq. 29) and 2a = 0.54 sqrt(R) is a genuine consistency between two columns of Table XI [read]. - Hidden assumption: the text never defines the "energy". Peek writes that the form of the rupturing energy "has not been considered as it was unnecessary" [read: p.234]. So no energy is ever computed and no units are attached to it. The word "energy" does not enter any equation. As a claim it is not a quantitative theory.

Independent check I did (not in the book): the form of the law follows from a streamer-type criterion. Suppose breakdown starts when the integral of an effective ionization coefficient alpha_eff (ionisations per cm, net of attachment) along the field line reaches a threshold K (about 18-20 in the Meek criterion, [memory, medium]) and suppose, near the threshold field E_o, alpha_eff rises linearly: alpha_eff = k (E - E_o). With E(x) = E_s r / x (E_s the surface field): integral from r to r_c of alpha_eff dx = k r E_o [ X ln X - X + 1 ], X = E_s/E_o, r_c = X r the radius where E falls to E_o. For small excess (X = 1 + eps) this is k r E_o eps^2 / 2, so eps = sqrt(2K / (k E_o r)), that is E_s = E_o (1 + c/sqrt r). The 1/sqrt(r) form drops out of the geometry (a 1/x field falling through the region where alpha_eff is positive) and the distance a = r eps is proportional to sqrt(r). With similarity scaling alpha_eff(E, delta) = delta phi(E/delta) (Townsend/Paschen scaling [memory, high]) and phi linear, k(E - delta E_o), the same steps give E_s = delta E_o (1 + c/sqrt(delta r)), exactly Peek's density form and his a = 0.301 sqrt(r/delta) [calc]. So the density law is not independent of the radius law; both come from the same similarity argument. Peek presented it as "found experimentally" [read: p.67]. Numerical test of the full (not small-eps) integral with E_o fixed at 29.8 kV/cm and one fitted constant C = K/(k E_o) = 0.028 cm, compared with Peek's law: -6% to +5% over r = 0.01 to 2 cm (e.g. r = 0.1 cm: Peek 58.2, model 54.9; r = 1 cm: 38.8 vs 37.2) [calc]. The fitted constant corresponds to alpha_eff of about 4/cm at 30 kV/cm and a slope of about 21 per cm per kV/cm for K = 18 [calc]; whether that matches measured ionization coefficients of air I have not verified (unknown). This is my model fitted to Peek's curve, not to data; it shows the structure, not the number 0.301.

b. Experiments. The distance itself is not measured. The only experimental handle is the dependence of onset on r, S, R, delta; see claim 1.

c. Tested? The idea (a finite ionising region) is what Hartmann 1984 tested and made quantitative using Meek's streamer criterion with humidity, pressure and temperature dependent coefficients [web: https://ieeexplore.ieee.org/document/4504655, abstract via the Investigator; not independently read]. Hartmann's inclusion of humidity in the ionization coefficient bears on Peek's "no humidity effect" statement; I have not read the result.

d. If true: nothing new; the later ionization/streamer picture contains it.

Verdict: split. The literal claim "air has constant strength g_o and energy must be stored over 0.301 sqrt(r) cm": Not testable as stated (an identity with a free function, energy never specified). Its mathematical content as "ionization over a finite region produces a g_o(1 + c/sqrt(r)) form": Plausible, untested by Peek's route, and consistent with mainstream breakdown theory by my own derivation above; the physical content was later supplied by others, so the original wording is Historically superseded but instructive.

Claim 3. Empirical corona-loss law p = (244/delta)(f + 25) sqrt(r/s) (e - e_o)^2 x 10^-5 kW/km

a. Internal consistency. - Units. The formula is empirical and not dimensionally homogeneous: the number 25 has units of Hz, 244 x 10^-5 carries kW per (km Hz kV^2), and the factors are fitted numbers. That is acceptable for an engineering fit if stated; the book gives the units (kW/km, kV effective to neutral, Hz) [read: p.237, 242]. Metric-to-English constants check (123.4 vs 123; 0.1889 vs 0.189; 392.7 vs 390) [calc, deep read]. - Structure. p = c^2 (e - e_o)^2 with c^2 = a (f+25), a = A sqrt(r/s)/delta. My refit of the printed c^2-versus-s/r table gives an exponent of -0.498 +/- 0.021 on s/r, so the sqrt(r/s) factor is strongly supported over 12 tests, and A = 243 [calc; Table LV, read: p.171]. The residual scatter is 7.4% rms, 16% worst. So the formula is good to roughly +/-10% over the fitted range (3 sizes of cable and wire, spacings 2-40 cm indoors and 250-310 cm outdoors), not better. - The 1/delta dependence is fitted too: c^2 is reduced "to 25 C and 76 cm" by multiplying by delta (Table LV; Table LIV shows c^2 drifting with temperature and pressure before correction) [read: p.170-171]. So delta is placed by a plot of c^2 against 1/delta (Fig. 125), not derived. - Quadratic law above e_o: the model says no loss below e_o. Peek states loss does occur below e_v from rough spots and "follows the probability curve p_1 = q exp(h - (e_o - e)^2)" with q, h undefined in the pages I read [read: p.240]. The formula is therefore a fit above the knee only; Peek says so [read: p.240]. - The worked 100-mile example and Tables LXXIX-LXXXII recompute correctly (e_o 52.0 kV; Table LXXX implies m_o = 0.87 uniformly); misprints: 3,300 for 4,300; one storm cell 8.0 for 7.3 [read: deep read; calc]. - e_o = 21.1 m_o r delta ln(s/r): g_o = 21.1 is 30/sqrt 2 = 21.2 (rms from peak), the book prints 21.2 on p.65 and 21.1 on p.68 and p.237 [read]. The apparent peak g_o for stranded cable is 25.6-25.8 kV/cm (versus 29-30 for smooth wire), i.e. m_o = 0.87 is just 25.7/29.5; Peek says so on the page where he states it [read: p.171-172]. Hence for stranded cable the "g_o" is an empirical surface constant that includes roughness, not the physical strength of air.

b. Consistency with experiments. The source experiments are Peek's own Lines A and B plus Watson's d.c. data and Hendricks's 1906 line. The Hendricks and Leadville comparisons in the book are by eye from plots. Au Sable: calculated 15 kW/mile against "preliminary rough measurements" of 15-20 kW/mile, with no spacing or weather inputs [read: p.247-248]. Peek's own caveat: practical-line measurements lack recorded temperature, barometer and end voltage [read]. Losses of this size, with 25% uncertainty on a measured value, are weak confirmation. Peek's 1927 paper (Part IV) re-measured the loss with a cathode-ray oscillograph and found the quadratic law above the visual voltage and "no loss until" it on polished wire [web: https://ieeexplore.ieee.org/document/5061442, abstract only, via the Investigator]; this is Peek confirming Peek, with a new instrument.

c. Tested? By Peek on his two experimental lines (the same data the constants were fitted to) and on lines he did not own (Leadville, Au Sable, Hendricks). I have no evidence in the record of the formula's replication by neutral parties beyond its continued use in later textbooks and articles (Electronics 1950, Westinghouse Engineer 1951, per the Investigator, snippets only). Not verified by me.

d. If true: nothing; this is an engineering correlation.

Verdict: Established as an empirical correlation inside its range (stranded and polished conductors, 25-60 Hz, fair weather, delta 0.6-1.1) to about +/-10% where the book's own data lie; weaker outside it (storm rule e_o x 0.8 and the low-voltage region are rules of thumb; Peek says so). "Corona will be an economic rather than a physical limit" is a forecast, not a mathematical claim (Not testable as stated here; the 1920s outcome is the Physicist's).

Claim 4. Strength of air at small gaps and in thin films

a. Internal consistency. - Table IX recomputes: (E/X) f with the printed E gives 195.7 kV/cm at X = 0.00254 cm (printed 196.0) [read: deep read; calc]. For a 25 micrometre gap the field is essentially uniform (gap << sphere radius), so the "gradient" is just V/d. - Cross-check between two separate tables of the appendix: Table IX's 0.497 kV peak at 0.00254 cm in air at atmospheric pressure corresponds to p x = 76 x 0.00254 = 0.193 cm Hg cm. The appendix Paschen table gives 395 V at 0.10 and 550 V at 0.30, which interpolates to about 470 V at 0.193 [calc]. Table IX says 497 V, 6% higher. This agreement is about as good as one could ask of tests with different apparatus, and it shows that Peek's "196 kV/cm" is simply Paschen's law at small pd, not an extra effect of the energy distance. Peek himself points to Paschen's law for these small spacings (footnote p.86). - The Paschen minimum: 350 V at p x = 0.05 cm Hg cm = 0.5 torr cm; the standard air values are 327 V at 0.567 torr cm [web: https://en.wikipedia.org/wiki/Paschen%27s_law, via search; confirms the standard air numbers]. Peek's table is 7% high in voltage and about 12% off in pd; his table's attribution is not given in the book [read: p.301]. - The sphere gradient 27.2(1 + 0.54/sqrt R): least-squares of Table XI gives 27.24 and 0.527; nine of eleven points within 1.7% [calc]. No fitting procedure printed [read]. - Air films in solids: g_a (entrapped film) and g_s (sphere gap of equal width) agree to a few percent down to 0.012 cm and then diverge at the thinnest films: 161 vs 196 kV/cm at 0.0025 cm [read: p.300]. The first-group temperature "285" is evidently 28.5 C: 285 C would give delta = 0.52, not the printed 0.96 [calc]. A glass-plate permittivity of 7.2-9.5 changes the field in the air film by the standard series-capacitor ratio; I did not recompute E_a to g_a. - Notation: "g_s" means film strength between metal spheres; the column headings are garbled in the text layer and need the page image (which Step 1 used).

b. Consistent with experiment? Yes: it reproduces what Paschen's law says (minimum near 330-350 V, rise at small and large pd) [web above]. The 25-micrometre value of nearly 200 kV/cm is a well-known feature of the left branch of the Paschen curve, but the book gives no uncertainty for a 0.001 inch gap set by counting screw turns from a bell-circuit contact [read: p.83; reader note in the deep read].

c. Tested? By Peek (spheres), Dubsky 1914 (air films, cited with "See" in the book, so authorship of the data is not stated) and, for Paschen, the classical literature [web above]. The Paschen table is unattributed in the book.

d. If true: nothing changes; this is Paschen-type behaviour.

Verdict: Established for the gap-dependence of air strength and the thin-film behaviour (Paschen's law; Peek's table agrees with the textbook minimum to about 7% in voltage). The "energy distance is cut short" explanation of the rise at small gaps is the same untestable identity as in claim 2 and adds nothing beyond Paschen.

Claim 5. Time lag: impulse voltages break insulation at higher voltage than 60 cycles

a. Internal consistency. - The tables are internally consistent to rounding, with small slips: needle ratios recomputed; oil ratio 390/170 = 2.29 vs 2.10 printed; varnished-cloth first row 2.32 vs 2.26; the 900 kc first needle spacing 1.65 cm is probably 0.65 [read: deep read; calc]. - The wave definition is given (wave 1: R = 520 ohm, C = 0.0005 microfarad, L = 0.312 mH; time to peak 0.5 microsecond; half-sine of about 1.2 microseconds half-period). Undamped LC frequency: 403 kHz [calc]; the labelled "frequency" of a single half sine of that duration is about 1/(2 x 1.24 microseconds). Wave 4 (long flat tail) is not a half-sine despite the label [read: p.302]. So the "55-900 kc" axis is a rough measure of pulse steepness (labels within 20% of the LC frequency), and any smooth curve of ratio against "frequency" has an x-error of that size. - The conclusion that spark-over needs "a minimum energy and hence a minimum time" is a hypothesis; the data are volt-time curves. A volt-time curve alone does not distinguish a minimum-energy law from other mechanisms (statistical formative lag, streamer growth time) [memory, high; standard high-voltage engineering]. The book does not attempt an energy balance, so there is no number for the "rupturing energy". - Measurement: the sphere gap is used as the impulse voltmeter, with its own impulse ratio about 1.00-1.06. There is therefore a circularity risk: needles are compared to a sphere gap that is itself a gap with time lag; the argument assumes the sphere's lag is small, which is supported only by the same table (sphere ratio within 5%) and the 25-cm spheres' large gaps. Peek warns "accuracy of impulse voltages best above 5 kV" [read: p.306]. The small-gap rows (impulse ratios 1.5-3.3 for spheres at sub-millimetre gaps, Table VI) are below that and should be discounted.

b. Consistency with experimental results. The qualitative effect (higher voltage needed for a very short pulse; ratio rising with steepness and falling density) is the volt-time characteristic of standard high-voltage engineering [memory, high]. Quantitative values from this book are Peek's alone.

c. Tested? By Peek himself with a GE impulse generator in 1915; the appendix is his own report [read: p.302], and the full paper is Trans. AIEE 1915 p.1857, which the Investigator located only as a citation [web: ETHW Transformers at Pittsfield part 2, via fetch]. The Investigator also records that Peek continued the work in lightning generators (3.6 MV in 1928, 5 MV in 1929, 10 MV in 1932) [web: https://ethw.org/Archives:Transformers_at_Pittsfield,_part_2, GE-history source, via fetch]. I found no neutral replication in the record I read. GE had a commercial interest (insulation and lightning protection of transmission equipment); the equipment ran by GE helpers, I flag that as an interested-party description of its own tests.

d. If true: nothing; it is standard transient breakdown physics.

Verdict: Established for the qualitative effect (accepted theory and Peek's experiments agree); the quantitative ratios (1.0-2.34 needle, 2-3.3 oil and solid, up to 5.8 at low density) are single-source (Peek, GE, 1915), uncorroborated in the record I read, and the "minimum energy" explanation is Not testable as stated.

Claim 6. D.c. versus a.c. breakdown

a. Internal consistency. - The book's statement "d.c. puncture voltage in air and oil is 1.41 x the a.c. effective" is the same as "d.c. = a.c. peak", for a sinusoid with rms V_rms and peak V_rms sqrt 2. The appendix tables use a.c. in kV max, so the ratio shown there is about 1.0; mixing the two conventions (the 1.41 vs the 1.0) is a trap for a reader, but both are right. My ratios [calc, read: p.295-299]: - Needles in air (Table I): d.c./a.c. peak 1.02, 1.01, 0.96, 0.93, 0.92 for 5-15 cm. - Needles in oil (Table VI): within 5%. - Concentric-cylinder corona (Tables II-III): d.c. = a.c. peak to about 1% except the finest wire (r = 0.0038 cm: d.c. 6.4 kV vs a.c. 4.9 kV). - D.c. spark-over of thin wires in cylinders: 1.2-1.5 x a.c. peak, negative wire higher than positive in every row. - Varnished cambric (Table IX): d.c./a.c.-peak 1.27 (1 sheet, 2 s) to 2.59 (4 sheets, infinite time) [calc]. - The attribution of the solid-insulation gap to a.c. dielectric heating is a hypothesis; the thickness dependence is also consistent with other mechanisms [memory, medium]; Peek gives no heating calculation. Table IX's "gradient" rows show d.c. gradient rising with thickness (60 to 80 kV/mm at infinite time), which is odd for a purely electrical strength and fits a thermal or time-dependent process. - Whitehead's compilation shows positive d.c. corona 4-6% below the a.c. peak, unlike Peek's own tables; no comment in the book [read: p.296].

b. Experiment. The d.c.-equals-peak result for air is a standard textbook statement [memory, high]. The solid insulation numbers are Peek's 1916 data, [read: p.295 heading, "Tests from Peek, A.I.E.E., June, 1916"].

c. Tested? Peek 1916; Watson (d.c. corona loss, J. IEE 1910; Table LII shows d.c. loss 1.5-1.7 x 60-cycle loss at the same maximum voltage on a fine wire, [read: p.169]) and Whitehead, Brown and Farwell on air. Replications by others are only the compilation in Table IV.

d. If true: nothing; a standard consequence of breakdown depending on the field maximum in air and on heating and time in solids.

Verdict: split. Air and oil (d.c. = a.c. peak, with the 8% and 20-50% exceptions in the book's own tables): Established. Solid insulation (d.c. 1.3-2.6 times a.c. peak, heating explanation): Plausible, untested as to cause (no controlled test of the heating hypothesis in the book); the data themselves are single-source (Peek 1916).

Investigator's findings, weighed and kept

  1. Fitch 1913 (Johns Hopkins; non-GE): tests over 5-108 cm Hg on nine conductor sizes; Peek's equation did not closely match, Fitch modified it, and reports support for the empirical relationship; CO2 results point to molecular spacing rather than absolute density [web: https://archive.org/details/influenceofdensi00fitcrich, via fetch tool, not seen by eye]. Bears on claims 1 and 2. This is evidence for the density law from a different lab and also a documented departure from Peek's original form; both stay. My point: the density form follows from Townsend similarity (claim 2), so Fitch's finding that molecular spacing, not mass density, matters (CO2 equal to air) is the expected behaviour of any theory in which alpha/p depends on E/p [memory, high]. Peek reports the same CO2 result without data [read: p.82].
  2. Hartmann 1984, "Theoretical Evaluation of Peek's Law": streamer-criterion derivation including humidity dependence [web: https://ieeexplore.ieee.org/document/4504655, abstract as recorded by the Investigator]. It supplies what Peek set aside as "unnecessary" (a physical mechanism). My derivation above is the structural reason such a calculation can reproduce the 1/sqrt(r) law; Hartmann's numerical agreement I have not seen.
  3. Peek 1927 Part IV (oscillograph measurements; quadratic law above the visual voltage; no loss on polished wire below it; probability law on cables; instantaneous critical voltage drops as voltage rises, so corona starts at the zero of the wave when e = 2 e_v) [web: https://ieeexplore.ieee.org/document/5061442, abstract via the Investigator]. This keeps claim 3 and gives a mechanism claim (the "probability law" the book calls undefined, with q and h unspecified in the pages I read). Not independent.
  4. Original papers 1911-1915 (Law of Corona I-III, Effect of Transient Voltages): these hold the full data, including the 9 points the book drops in the fit, and the full impulse report. A reader or a later reviewer could redo my discard test directly from them. I did not obtain them.
  5. Third edition 1929 may alter the energy-distance wording and loss law; not read.
  6. Peek's biography and GE setting (GE employee, Chief Engineer of Pittsfield 1931, death 1933; ETHW gives 1931 vs 1933 for the death inside one source) [web: https://ethw.org/Archives:Transformers_at_Pittsfield,_part_1, part_2, via fetch]: GE is an interested party in the measurements (see claims 3 and 5) but the formulas were adopted by others for decades. "No silence in the record": his work continued to 1933, so there is no absence-of-evidence anomaly to analyse.
  7. Whitehead 1910/1911/1917, Watson, Dean, Dubsky, Scott and Mershon (1898), Faccioli, Hendricks: named in the book as outside data; the Investigator found Whitehead's 1910 paper only as a citation. Kept as leads.
  8. The Investigator's blocked or unreached items (Union College 84 items, Chronicling America, NYT, Espacenet, Stanford) are not mathematical and I did not weigh them.

Web checks (budget 6, 2 used)

  • Paschen minimum for air 327 V at 0.567 torr cm: confirmed [web: https://en.wikipedia.org/wiki/Paschen%27s_law, via search result].
  • Hartmann 1984 abstract: my fetch of the IEEE page returned no content; I rely on the Investigator's tagged abstract and say so.
  • Not checked: the Meek criterion constant K (about 18-20) and the numerical alpha_eff of air; the results of my streamer toy-model depend on them only for the physical magnitude, not for its form.

What would settle open mathematical points

  • Re-fit the 1911 original data with all points and the stated criterion for discards (claim 1).
  • Compare the streamer-criterion calculation with measured alpha_eff for air and with Peek's curve (claim 2).
  • Obtain Fitch's modified formula (claim 1).
  • Read pp.154-188 chapter V for the full 109,500 cm Line A fit including the 12-test c^2 table in the original scan (claim 3).