Peek-Frank: Physicist review (Step 2)

Reviewer: PhD physicist. Source reviewed: F. W. Peek Jr., Dielectric Phenomena in High Voltage Engineering, 2nd ed., McGraw-Hill, 1920 (325-page scan, OCR). Peek worked for General Electric (GE), a maker of transmission equipment; that is an interest, and GE's Consulting Engineering Department did the experiments [read: preface, facts.json].

Tags: [read] from the extracted text or deep reads; [web] with URL; [memory] with confidence; [calc] my own arithmetic (numpy venv).

Source gap carried forward. The text is OCR on 324 of 325 pages. Chapter III, Chapters VIII-IX and the appendix pp.295-311 were read from page images by the deep reader. Chapter V (the Line A / Line B loss experiments, pp.154-188) and the sphere-gap chapter were digest-only; I read some of that text myself (see "Corrections"). The Investigator downloaded nothing (research/files/ does not exist), so the original 1911-1915 papers, the 1927 Part IV paper and Fitch's thesis were not read by anyone; they are leads only. Every conclusion below rests on the book plus the cited web sources, not on [memory] alone.

0. Physics you need first (plain language)

• Breakdown of air. A free electron in a strong electric field gains energy between collisions; if it gains enough it knocks another electron off a molecule ("impact ionization", Townsend 1900s). Each electron starts an avalanche. Air also contains oxygen, which captures electrons ("attachment"). The field at which ionization and attachment balance is about 110 Td (Td = townsend, a unit of field divided by gas number density) [web: https://arxiv.org/pdf/physics/0508109 and search result on critical reduced field]. At 25 C and 1 atm, 110 Td is about 27 kV/cm [calc: 110e-17 V cm^2 x 2.45e19 cm^-3]. Peek's "strength of air" g_o = 30 kV/cm peak (21.1 kV/cm rms; 30/sqrt 2 = 21.2) is the same number to within a few percent. So g_o is not arbitrary: it is roughly where net ionization begins.
• Corona. Around a thin wire the field falls off as 1/distance, so the field is large only in a thin sheath. Air glows ("corona") when an avalanche in that sheath grows to roughly 10^8 electrons (about 18-20 e-folds of growth), the "Meek/Raether streamer criterion": the integral of the effective ionization coefficient along the path equals about 18-20 [web: https://arxiv.org/pdf/physics/0508109]. A thinner wire has a thinner high-field sheath, so the surface field must be higher before the integral reaches the threshold. That is Peek's observation.
• Paschen's law. Breakdown voltage of a uniform gap depends on pressure times gap (p x), with a minimum near 330 V in air [memory, high; consistent with Peek's own appendix table, see claim 4].

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

a. Mathematics. Internally consistent as an equation. Units: 0.301 must carry cm^(1/2) (r in cm); Peek never says so [read: deep, p.72]. The fit arithmetic (Sigma-Delta group averages, Table IV: slope 8.99, intercept 29.84, ratio 0.3013) is correct [read: deep, p.75]. Reconstructed from the printed data: 10 of 19 points kept, 9 discarded with no stated criterion; kept and discarded points have overlapping residuals (kept -5.5% to +4.0%, discarded -6.2% to +4.3%). An ordinary least-squares line through all 19 gives g_o = 30.4, slope 8.71 (ratio 0.286) [calc, deep]. Does this matter? For a 0.02 cm diameter wire (r = 0.01 cm) Peek's constants give 119.8 kV/cm and the least-squares ones 117.5 kV/cm [calc], 2% apart, below the scatter. The discarding is a method flaw that does not change the practical result. Rms deviation of the law over all 19 Table III points is 3.5%, maximum 6.2% [read: deep, calc]. Table VI-VII fit to under 1.7% [deep], but those tables were part of the data fitted, so that is in-sample agreement. Table I (spacing independence) averages 98.4 not the printed 99.0, a trivial slip; it shows g_v flat (96-100 kV/cm) while spacing varies 42-fold, a strong result [read: p.70, deep].

b. Consistent with experiment? Yes, for the conditions of the tests: smooth polished wires, 60 Hz, around 1 atm, delta from 0.07 to 1.3, r about 0.01 to 2.5 cm. Specific results: - Spacing independence (Table I), material independence (tungsten, copper, aluminium, iron interleaved on one curve, Table II) [read: p.70-72]. - Density: Table VIII is within 2% for delta 0.146-1.0 but the two lowest-density points (delta 0.069 and 0.141) are 7.8% and 7.7% below the formula; the book does not comment [calc, deep]. Peek's claim "holds down to delta = 0.02" is therefore only loosely supported in the printed table (lowest tabulated delta 0.069). - My own shape check [calc]: I integrated a crude single-exponential Townsend coefficient (A = 15 /cm/torr, B = 365 V/cm/torr, no attachment) along the 1/x wire field and solved for the surface field giving a growth of K = 10-18. The predicted rise from r = 1 cm to r = 0.01 cm is a factor 2.9-3.3, against Peek's 3.08 (30(1+0.301/0.1) / 30(1+0.301/1) = 120.3/39.0). So the shape of Peek's law follows from ionization physics. The absolute values from my crude model are 30-50% too high (for example 58 kV/cm vs Peek 42.8 at r = 0.5 cm for K = 18); the crude coefficients ignore attachment, which lowers the net ionization near 30 kV/cm, so this shows shape agreement only, not a derivation. - Independent measurement: Fitch (Johns Hopkins, Whitehead's laboratory, 1913) measured critical corona intensity at 5-108 cm Hg on nine conductor sizes. As returned by the fetch tool (not checked by eye by the Investigator), Peek's equation "didn't closely match", Fitch modified it, the modified form fitted "about as closely as the readings can be taken", and the thesis concludes the results support Peek's relationship; CO2 tests suggest molecular spacing, not absolute density, matters [web: https://archive.org/details/influenceofdensi00fitcrich]. This is a non-GE test that partly corroborates and partly modifies the density term. I did not read the page images; the exact modified formula is unknown. - Later theory: Hartmann 1984 derives the onset gradient from Meek's streamer criterion with an apparent ionization coefficient depending on humidity, pressure and temperature, and compares with "a variety of experimental results" [web: https://ieeexplore.ieee.org/document/4504655]. Protsenko and Trotsenko 2014 is a centenary review ("dedicated to the memory of ... Frank William Peek") [web: https://kpi.ua/nv-abstract-14-6]; not read beyond the citation. Peek's law is still used for engineering estimates of corona onset; a "modified Peek's law" is still used for corona thresholds [web: https://typeset.io/topics/peek-s-law-2vkw38v0, search snippet]. - Where the book is weaker. Humidity independence ("Humidity has no effect") is asserted with no humidity data in pp.60-92 [read: deep]. Hartmann's inclusion of humidity in the ionization coefficient and a reported dependence of corona loss on relative humidity near inception [web: search snippet, https://digital-library.theiet.org/content/journals/10.1049/piee.1971.0131] point the other way, though I did not read those sources in full. Treat the humidity statement as untested in this book, and probably only approximately true (small effect on onset). Oil and water: the detailed section gives water g_v about 9(1+0.815/sqrt r), roughly half of dry, and oil 3-7% below dry, which sits loosely with the summary "oil has very little effect" [read: deep, p.91-92]. - Sphere/cylinder constants differ (31, 30, 27.2 for balanced cylinders, parallel wires, spheres). Peek attributes the spread to field "balance" [read: p.88]. A physicist reads this as a sign that g_o is a fit parameter of the geometry, not a pure property of air.

c. Tested? By Peek and GE helpers (H. K. Humphrey thanked), on his own tests; independently by Fitch 1913 (above) and Watson (d.c. data used in Table V, which fall 2-11% below a.c. peak, so d.c. polarity matters) [read: p.77]. Whitehead 1910-11 programme ran in parallel [web via Investigator]. Not replicated in this review. Widely used since 1911, which is a form of long field testing but not a controlled test.

d. If true, what changes? Nothing. The law is an engineering fit that mainstream ionization physics already explains in shape.

Verdict: Established (empirical onset law for smooth wires, ac, near 1 atm, within about 5%). Split: the humidity-independence part is Never properly tested in the book (asserted without data); the extension to delta = 0.02 is Plausible, untested (the two lowest-density points are 8% low).

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

a. Mathematics. Internally consistent but not a derivation. Eqs. (21)-(23) leave a = phi(r) free; dividing (21) by (22) gives a = r(g_v/g_o - 1), so any measured g_v(r) curve can be restated as "constant g_o at a distance a(r)". The text's "the experimental values bear out the above theory" is a restatement of the fit [read: deep, p.72; my verification: g_v = e/(r ln(s/r)) and the field at distance a is g_v r/(r+a), so g_v/g_o = 1 + a/r, and a = 0.301 sqrt r gives Peek's law exactly; the algebra is trivially true]. What can fail is that one g_o fits all r, s and R, and that a(r) takes the simple square-root form; both held. Notation clash: "a" is the coefficient in (24) and the distance two lines later [read: deep]. Peek writes the density form as a = 0.301 sqrt(r/delta), "found experimentally" [read: p.67], which is a fit too, the theory predicting only some phi(delta). Peek's own wording: "Thus far the form of this rupturing energy has not been considered as it was unnecessary" [read: p.234]. The 0.301 sqrt r distance also is "many times greater than the mean free path" [read: p.235].

b. Consistent with experiment? The qualitative content is borne out. The same distance concept predicts that gaps shorter than a give higher strength; sphere-gap data show the gradient rising for X < 0.54 sqrt(R), and the independently fitted constants agree (X = 0.54 sqrt R from column 2 of Table XI; alpha = 0.54 from column 3, the factor 2 in a = 0.27 sqrt R) [read: deep, p.87-88]. That cross-check is genuine. Peek's footnote cites Bergen Davis's 1914 "Theory of Corona" for an ionic explanation. In modern terms the "energy-storage distance" is the length of path over which the field exceeds the critical value and the avalanche grows (above); the streamer criterion replaces Peek's unspecified "energy" with a count of avalanche growth. Peek's "ionic saturation" is not the modern picture, and the proposed link between 0.301 and mean free path is not tested.

c. Tested? The concept was not tested separately from the fit; Hartmann 1984 (above) is a later calculation by others that reproduces onset gradients from ionization physics. Whether Peek's 1927 Part IV paper ("The Mechanism of Corona Formation and Loss", Trans. AIEE 46:1009-1024) specifies the energy is unknown; only the abstract was read (cathode-ray oscillograph; loss quadratic above the visual critical voltage; no loss on polished wires below it; corona starts at the voltage zero when applied voltage is twice the visual critical value) [web: https://ieeexplore.ieee.org/document/5061442]. Highest-priority read for the next reviewer.

d. If true, what changes? Nothing in mainstream physics; it was an early, fair version of the streamer criterion. The "energy" language belongs to the pre-1920 picture.

Verdict: Historically superseded but instructive (as a mechanism; replaced by Townsend-Meek-Raether avalanche-to-streamer theory). Split: the algebraic restatement "constant strength at a distance" is Not testable as stated (true of any curve).

Claim 3. Empirical corona-loss law p = (244/delta)(f+25) sqrt(r/s)(e - e_o)^2 x 10^-5 kW/km; e_o = 21.1 m_o r delta ln(s/r) kV; storm rule e_o x 0.8; corona an economic, not physical, limit

a. Mathematics. Dimensionally the constants hide units (it is a fit in kV, cm, Hz, kW/km). The English constant 390 matches 244 x 1.609 = 392.7 and 123 = 21.1 x 2.54 x ln 10 = 123.4 [calc, deep]. The 100-mile worked example recomputes: delta = 0.924, e_o = 51.9-52.0 kV, loss coefficient 0.0424, visual voltages 62.6 and 71.3 kV to neutral [calc, deep; I re-ran e_o and got 51.90 with unrounded delta]. Misprints: A = 241 on p.237 vs 244 on p.242 (1.2%); "0.80 per cent" for 80%; one Table LXXIX cell 8.0 vs 7.3 computed (9%), "3,300" for 4,300; Hendricks line 48 in printed as 132 cm; "critical voltage may also be decreased by grouping conductors", probably a misprint for increased [read: deep]. None changes the central result. The (f+25) form (a loss proportional to frequency plus a constant) and the sqrt(r/s) dependence are fitted by group averaging, not derived. Peek himself notes the loss below e_v "actually follows the probability curve" [read: p.240].

b. Consistent with experiment? With his own Line A (3/0 cable, 310 cm spacing, 109,500 cm conductor length) and field comparisons, as reported: Leadville, 1/0 weathered cable at 60 cm Hg, calculated vs measured (Faccioli) "agreement is very interesting" but the numbers are only plotted (Fig. 173); Au Sable 125-mile line, calculated 15 kW/mile vs "preliminary rough measurements" 15-20 kW/mile; Hendricks 1906, non-sinusoidal wave "an exact agreement could not be expected" [read: p.247-248]. Peek is candid that data are incomplete (temperature, barometer, voltage rise unrecorded). Note: Peek's 1927 paper abstract reports the quadratic law re-measured with an oscillograph to 1% accuracy at 0.1 W [web: https://ieeexplore.ieee.org/document/5061442]. - Closure of an open point from Step 1 (see Corrections): Line A was single-phase [read: p.xv "Tests were made on single-phase lines unless otherwise noted"; also p.287-area text "As these measurements were made on a single-phase line, the voltages between wires were twice the value given"]. Then e_o for Line A = 70.0 kV to neutral (m_o = 0.87, delta = 1.032), i.e. 140 kV between wires, matching the 145-150 kV read from Fig. 167 [calc, deep]. The loss law agrees with its own fitting line, as expected. - "Corona rarely a limit; 250 kV needs no excessive conductor": this was a reasonable forecast for 1920. Later EHV lines (above 400 kV) were limited by corona noise and loss, and bundled conductors solved it [memory, medium; not web-checked]. The 1920 statement is superseded in the sense that corona did become a design constraint, but the remedy (bigger effective radius) follows Peek's own e_o formula.

c. Tested? By Peek (fit), by Faccioli and Hendricks (GE and Colorado data used by Peek), Au Sable measurements; Scott and Mershon 1898 Colorado cited. Continued use in engineering texts and 1950-51 articles (Electronics 1950, Westinghouse Engineer 1951) [web via Investigator: worldradiohistory.com]. No controlled neutral replication in the sources I have.

d. If true, what changes? Nothing; it is an engineering correlation, not a law of nature.

Verdict: Established as an empirical fair-weather engineering estimate for stranded conductors in the fitted range. Split: the storm rule (e_o x 0.8) and the field comparisons (Au Sable) are Plausible, untested in this book (single rough comparison, no numbers for Leadville); the claim that corona is only an economic limit is Historically superseded but instructive.

Claim 4. Strength of air at small gaps; entrapped air films; Paschen table

a. Mathematics. Table IX arithmetic reproduces (195.7 vs 196.0 kV/cm at 0.00254 cm) [calc, deep]. Peek's approximation (30) for sphere fields is accurate "only when a = 0" (his words); the exact formula (26) by Dean is used [read: deep, p.88]. One misprint: temperature "285" for the glass-plate table, which must be 28.5 C [calc, deep].

b. Consistent with experiment? Yes. Paschen's law with Peek's own appendix table: minimum about 350 V at p x = 0.05 cm Hg cm (0.5 torr cm) vs the textbook 330 V near 0.5-0.6 torr cm [memory, high; reader's comparison in deep]. My Townsend-form check [calc]: uniform gap pd = 1.9 torr cm (0.0025 cm at 1 atm) gives about 380 V; Peek's 0.0025 cm sphere gap sparked at 363 V rms = 513 V peak, 35% higher, and the sphere field is non-uniform; gap setting by counting screw turns from a bell contact at 0.0025 cm has large uncertainty (not stated). Right order of magnitude. A uniform-field empirical formula for air, V = 24.22 pd + 6.08 sqrt(pd) kV (pd in cm at 1 atm) [memory, medium], gives 0.36 kV at 0.0025 cm, 30.3 kV at 1 cm and 26.1 kV/cm average at 10 cm, so "30 kV/cm" is right near 1 cm and falls by about 15% at 10 cm. Peek's claim that g_o is a constant for planes is an approximation.

c. Tested? Paschen 1889 and later work; Dubsky 1914 for air films [read: appendix]. Peek's films data show entrapped films (about 0.01 cm and thicker) break down at the gradient of an equal free gap (g_a 53 vs g_s 50.6 at 0.056 cm), thinnest films lower (161 vs 196 at 0.0025 cm) [read: p.300]. The practical consequence (voids in insulation produce partial discharge) is standard engineering knowledge [memory, high].

d. If true, what changes? Nothing.

Verdict: Established. The "energy distance cut short" explanation is qualitatively the left-hand branch of Paschen's curve (too little path to build an avalanche); superseded in mechanism, not in data.

Claim 5. Impulse voltages break insulation at far higher voltage than 60 Hz ("dielectric spark lag"); sphere gaps read impulses; rain barely affects impulse flashover

a. Mathematics. Peek's pulses are single half-sine waves; I checked the stated frequencies against the printed L and C: 1/(2 pi sqrt(LC)) gives 45, 101, 285, 781 kHz for the "55, 100, 350, 900 kc" blocks (the labels are round, 20% off except 100 kc) and 403 kHz for wave No. 1; half-period at 900 kc is 0.56 microseconds and at 55 kc 9.1 microseconds [calc, deep]. One apparent misprint (needle spacing 1.65 cm at 900 kc). The tables are internally consistent: the same impulse sparks the sphere gap and the needle gap, so the sphere gap works as a voltmeter [read: p.302-303].

b. Consistent with experiment? Yes. Needle-gap impulse ratio: 1.00 at 55 kc for short gaps, 1.12 at about 13-14 cm, 1.36 at 100 kc, 1.51 at 350 kc, 2.34 at 900 kc [read: p.303, deep]. Oil 2-3.3, solids 2.2-2.5, up to 5.8 at low air density [read: deep, p.305-306]. The modern account: breakdown needs both a statistical lag (waiting for a starting electron) and a formative lag (time for the avalanche and streamer to bridge the gap), so the voltage needed rises as the pulse shortens [memory, high]. Peek's "minimum energy" language is replaced, but the time-dependence is correct. Sphere gaps remain an international standard for measuring peak impulse voltage, accuracy about +/-3% for gap spacing S <= 0.5D [web: https://iteh.es/catalog/standards/sist/c2e5bb77-cee9-434c-8385-19e10adef442/sist-en-60052-2004 and search results on IEC 60052]; this matches Peek's 0-6% (mostly 0-3%) finding [read: p.302-303]. Rain lowers 60 Hz insulator flashover 36-40% and impulse flashover 2-3% [read: p.307-308]: the physics (water film and drops change the surface field and creepage over seconds, but there is no time for it on a microsecond pulse) is plausible [memory, medium].

c. Tested? By Peek (1915 AIEE paper, GE) with one generator (circuit in Fig. 147a). I did not find a source for an independent replication in the project sources; the later adoption of sphere gaps in standards implies the sphere result was confirmed, but I cite no replication of the needle and oil ratios. His work continued into 3.6 MV (1928), 5 MV (1929) and 10 MV (1932) lightning generators [web via Investigator: ethw.org, a GE-history source; and Wireless Engineer Oct 1932 abstract].

d. If true, what changes? Nothing. This is the book's most durable contribution.

Verdict: Established (time lag, impulse ratio, sphere-gap voltmeter). The "minimum energy" interpretation is Historically superseded but instructive.

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

a. Mathematics. The factor 1.41 is sqrt 2, the peak-to-rms ratio of a sine wave. Peek's tables are consistent with this; the exceptions are visible in the book's own numbers: thin wires in cylinders (d.c. above a.c. peak), long needle gaps about 8% below, Whitehead's 1917 compilation shows positive d.c. corona 4-6% below the a.c. gradient, and Watson's d.c. corona data in Table V fall 2-11% below a.c. [read: deep, p.77, p.295-299].

b. Consistent with experiment? In air and oil, d.c. breakdown equals a.c. peak in uniform-field conditions [memory, high]. In solid insulation, d.c. strength exceeds a.c. peak (Peek: varnished cambric 1.27 to 2.59 times, rising with thickness and time) [read: deep, appendix Tables VII-IX]. Modern accounts attribute the gap to dielectric heating (loss in a.c.), and partial discharge in voids [memory, medium]. Peek's explanation is heating alone.

c. Tested? By Peek in 1916 (GE), and by Whitehead and Watson independently for corona; the differences between the a.c. and d.c. corona values in these independent tests (4-11%) are larger than the within-lab scatter, so "equal to a.c. peak" is a first approximation.

d. If true, what changes? Nothing.

Verdict: Established (air and oil, with a few-percent polarity exceptions); Plausible, untested in this book for the heating-only explanation of the solid-insulation excess (no separate heating measurement shown).

Investigator's findings: weighed and kept

Retained in the record whatever the verdict: - Own later work: Peek 1927 Part IV (oscillograph; relevance above); original papers Part I 1911 (77 pp., free on archive.org), Part II 1912, Part III 1913, sphere gap 1914, altitude 1914, transient voltages 1915; third edition 1929 and 1926 "Lightning and other high-voltage phenomena"; 19 US patents 1908-1932 and CA275641 (GE, assigned; GE is the interested party). Part I holds the full data including the points the book discards. Bears on claims 1-2 and 5 [web: Investigator, sites.md]. - Supporting: Fitch 1913 thesis; recognition (APS Fellow 1922, Rowland Prize 1923, Levy Medal 1926); continued citation in the 1950-51 trade press [web: Wikipedia, worldradiohistory]. Interested-party note: the recognition is from professional bodies in the engineering community Peek belonged to; it is not an independent test. - Related: Hartmann 1984, Protsenko and Trotsenko 2014, Whitehead 1910. - Historical record: GE test man 1905, Pittsfield from 1916, Chief Engineer 1931; declined a Stanford chair 1931; death 27 July 1933 near Gaspe, Quebec (train and car), with ETHW internally inconsistent (1931 vs 1933). The "Wrestling with Nature" snippet ("In 1916 Charles Steinmetz argued that F. W. Peek Jr.'s experiments on high-voltage lines ...") is truncated and unread; it may show how GE presented the corona work. The record does not go dark: papers to 1927, lightning work to 1933, so no absence-of-evidence analysis is needed. Open: whether GE after 1933 carried the "energy distance" interpretation forward.

Corrections to Step 1

1. Line A phase is not open. claims.md says it is "unstated"; the book's general note says "Tests were made on single-phase lines unless otherwise noted" (printed Table of Symbols page, PDF p.20) and the loss-data caption for the outdoor tests says "As these measurements were made on a single-phase line, the voltages between wires were twice the value given" [read: extracted text lines near p.xv and p.287]. With that, e_o = 70.0 kV to neutral matches Fig. 167's 145-150 kV between wires (140 computed) [calc]. Step 1 should drop the request for a deep read on this point; a deep read of Chapter V is still worth doing for the fitting data (not done).
2. The sphere constant. claims.md gives 27.23(1+0.54/sqrt(delta R)) for the "sphere-gap average"; in Chapter III Peek prints 27.2 (eq. 28), and 27.23 appears in the later density-correction passage (p. ~127-131 text) [read]. Both are in the book; they are the same fit stated twice. Not an error.
3. "9 of 19 points discarded" (claims.md) vs "10 of 19 kept" (facts.json) are consistent, but neither says the criterion is unstated and the discarded set is inferred, not printed [read: deep, p.74]. The effect on g_v is 2% at r = 0.01 cm [calc].
4. Humidity. facts.json lists "Humidity has no effect" as a key claim without noting that no humidity data appear in pp.60-92 [read: deep]. I rate it Never properly tested in the book.
5. "Down to delta = 0.02" is repeated in facts.json as a result; the printed tables go no lower than delta = 0.069, and the two lowest are about 8% below the formula [read: deep, Table VIII]. Not an error in the book's text, but facts.json should say it is asserted.
6. Claim 2 wording. Step 1 says Peek "does not derive 0.301" and the deep read says the reinterpretation is algebra; both correct. I add that the algebra is a tautology for any curve (Claim 2a).
7. facts.json "mainstream_conflict" says bundling "conflicts with the earlier-printed result (p.95-98 digest: split conductors give 16-30 percent greater critical voltage)". I did not verify this against the page text; the digest line is a secondary summary. I agree with the direction (bundling raises onset voltage [memory, high]).
8. No errors found in the Chapter VIII-IX arithmetic; my recomputation of e_o = 51.9-52.0 kV agrees.

Verdict summary

Web calls used: 3 (of about 6). Web checks that failed or were thin: Hartmann and Fitch were not read beyond the Investigator's abstracts; the claim that humidity affects corona onset rests on search snippets, flagged above.