Mathematician review: Schuster and Hemsalech, "The Constitution of the Electric Spark" (1899)

Role: PhD mathematician (Step 2). I checked derivations, units, notation and hidden assumptions. Sources: the extracted text of the 6-page abstract [read], the deep read (which I re-checked against the text), and the Investigator's leads in sites.md [web, as the Investigator tagged them; I did not re-fetch them]. Tags: [read] document, [calc] my arithmetic, [web] URL, [memory] with confidence. No source gap: the one document (abstract, Proc. Roy. Soc., pp. 331-336) is fully readable. The full paper (Phil. Trans. A 193, 1900) is NOT in the project, so the tables for Al, Mg, Bi, Cd, Hg are unseen. Where a conclusion rests on a single sentence of the abstract I say so.

Mathematics you need first (plain language)

• Moving-film method. A spectrograph slit is imaged on film that moves at speed u (about 100 m/s here). A spectral line is a thin image of the slit; if the emitting atoms sit still, the line is drawn out along the film's motion but stays straight. If the emitters move along the slit direction (from electrode toward the gap centre) at speed V, the line is tilted. Time maps to film position by x = u t, and position along the slit is y = V t, so the slope is dy/dx = V/u, hence V = u (dy/dx) (unit magnification, which the authors state: slit to electrode distance = 4 x focal length) [read: p.2; calc].
• Worked example. Zinc, V about 500 m/s, u = 100 m/s: slope 5. Travelling 2 mm along the gap (the V1 definition) takes 2 mm / 500 m/s = 4 microseconds, during which the film moves 0.4 mm. So the line image is a 2 mm rise over a 0.4 mm run. A reading error of 0.02 mm in that run is 5 percent in V; the authors themselves say "a very slight error in measurement will also affect the results considerably when the total displacement measured is small" [read: p.2]. Fast metals (V = 1420 m/s) give a run of only 0.14 mm, so the same absolute error is about 14 percent [calc].
• Speed of sound in an ideal gas: c = sqrt(gamma R T / M), gamma the ratio of specific heats, R = 8.314 J/(mol K), M molar mass.

Verification of Step 1 (and corrections)

Corrections to Step 1

1. facts.json says the sound-speed formula was "reconstructed by a numerical check, not from a page image". The deep read did render p.3 and p.5 and confirmed the formula and Table I. I also read the text layer (V = 80 sqrt(T/rho), T = 2700, rho = 56) [read: p.5]. Correct: formula confirmed from the page image per the deep read; facts.json how_read understates what was done.
2. facts.json lists the "zinc 4925 vs 4811" claim neutrally. The abstract contains an internal wording slip that Step 1 missed: p.3 says the "uniformly higher velocity deduced from the double line 4925", but Table I shows 4811 higher than 4925 in every one of the nine (gap, jar) cells (recomputed below). Recomputed [calc]: 1014/814 = 1.25; 668/556 = 1.20; 529/416 = 1.27; 501/400 = 1.25; 548/499 = 1.10; 545/415 = 1.31; 1210/723 = 1.67; 1526/1061 = 1.44; 492/435 = 1.13. So in all nine cells 4811 > 4925, by 10 to 67 percent. The next sentence of the paper ("smaller apparent velocity from 4925") agrees with the table. The deep read caught this; facts.json did not carry it. I take "higher" on p.3 as the authors' slip (their cross-wire explanation on p.4 makes the 4925 doublet read LOW).
3. Step 1 states cadmium crosses the gap "in about 6e-6 s" and speeds "roughly 400-1500 m/s" together with "560 m/s for cadmium". These are not mutually consistent (see claim 1 below): 5 mm in 6 microseconds is about 830 m/s, not 560 m/s. Step 1 did not flag it; the deep read did.
4. claims.md item 4 says "implying the molecule in the spark is not much lighter than the normal vapour molecule". This is the authors' conclusion, but it is nearly empty as an inference (see claim 4). Step 1 reports it as a result; I classify it as a weak inference.
5. claims.md item 5 says "no simple relation to atomic weight". The Investigator found that Royds (1908), from the same lab, summarised the 1899 result as "the lighter metals had a greater velocity than the heavier metals" [web: https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.1908.0023, via Investigator]. The abstract's own words are "no simple law" and it lists aluminium over 3x zinc, bismuth anomalous. These can both be true (a trend with exceptions); a later summary is not what the abstract says.

Notation and units audit

• Velocities in "metres/second"; wavelengths 4925, 4811 unlabelled but angstroms by convention [memory: high]; text gives 4924.8 and 4810.7 [read: p.2].
• The "annular space of 10.8 cm" is the difference of diameters 33 - 22.2 = 10.8 cm, so the exposed band is 5.4 cm wide radially. Mean radius about 13.8 cm; 2 pi (0.138)(120) = 104 m/s, matching "about 100 m/s" [calc]. Film speed varies across the band by about 40 percent (radius 11.1 to 16.5 cm) [calc]: speed at the inner edge about 84 m/s, outer about 124 m/s. The abstract does not say that this gradient was corrected for; the slope-to-velocity conversion needs the local speed. Unknown from the abstract.
• "rho = vapour density referred to hydrogen": for rho = 56 this means density relative to H2 = 2 g/mol, so the molar mass is 112 g/mol, atomic cadmium (112.4) [calc; memory: high]. The notation "p" in the text layer is rho.
• Capacity 0.033 microfarad over 6 jars is about 0.0055 microfarad each if equal [calc].
• Cap/period check: T = 2 pi sqrt(LC) = 2 microseconds with C = 0.033 microfarad gives L = T^2/(4 pi^2 C) = 3.1 microhenry [calc]; plausible for a short low-inductance circuit; not stated in the source.

Claim-by-claim (questions a-d)

Claim 1: metal vapour crosses the gap measurably (about 6e-6 s to the centre for cadmium; roughly 400-1500 m/s)

a. Mathematically consistent? Partly. The inference chain (film speed -> time -> slope -> speed) is sound. But the numbers do not close: 6e-6 s over 5 mm (half the gap) is 830 m/s; removing the 0.5 microsecond of air luminosity gives 5.5 microseconds and 910 m/s. The authors give 560 m/s as cadmium's "average molecular velocity found" [read: p.5], which would take about 9 microseconds [calc]. A ratio of about 1.5. The abstract does not reconcile this; possible readings (V1 over the first 2 mm only vs. later speed, or loose rounding) are my guesses. Also zinc V1 about 500 m/s over 2 mm = 4 microseconds, so 6 microseconds for 5 mm for Cd needs speeds beyond 2 mm of about 1000 m/s. Not impossible (the authors say velocities near the pole are uncertain), but it is not demonstrated. Table I internals: the zinc values at 1.54 cm range 435 to 1526 m/s (a factor of 3.5) for what should be the same quantity; the authors mark the 6-jar values "?" and say "no certain conclusions can be drawn" [read: p.4]. There are no error bars or numbers of photographs in the abstract. b. Consistent with established experiments? An order-of-magnitude speed of a few hundred m/s to a km/s is the same as vapour from a hot electrode expanding into hot gas (speed of sound in 2700 K metal vapour is about 580 m/s [calc]). It is consistent with later observations of electrode-vapour expansion in vacuum-arc and spark research [web, via Investigator: Beilis 2018 IEEE; Monnin 2022 HAL thesis]. Against: Hull (1906) Doppler measurement with Cd/Hg amalgam electrodes found no shift above 0.002 angstrom [web: https://archive.org/stream/proceedingsroya02britgoog/proceedingsroya02britgoog_djvu.txt, via Investigator]. Check [calc]: a shift of 0.002 angstrom at 4000-4700 angstrom corresponds to v = c (0.002/lambda) = about 130-150 m/s, matching Hull's stated bound of 150 m/s. A source moving at 500 m/s toward the observer would give 0.008 angstrom at 4700, four times larger. So Hull's result contradicts moving emitters IF his line of sight had a component along the velocity; the Investigator flags that the geometry (observing along or across the spark) needs checking. I have not read Hull's paper; this stays an open conflict. c. Tested/replicated? Replicated in method by Royds in Schuster's lab (1908, 1910) with a velocity table by metal [web, via Investigator]; Schenck 1901, Milner, Strutt 1913 follow-ups cited [web, via Investigator], not read. Challenged by Hull 1906 (Doppler) and by Royds' own point (2): Schuster and Hemsalech "did not discuss any possibilities of the propagation of luminosity through vapour which may be, as a whole, stationary" [web, via Investigator]. So there are replications of the slope but a competing reading of what the slope means. d. If true, what changes? Nothing in mainstream physics. It is an ordinary kinematic measurement. If the Hull reading is correct instead, the quantity measured is the speed of a luminosity front, not of matter.

Hidden assumption (key): the slope V = u dy/dx equals the matter velocity only if the emitting atoms are carried along. Mathematically, a line's slope measures the speed of any front of emission, matter or excitation. A front of excitation (current-carrying luminosity wave) obeys the same kinematics. The data alone cannot separate them; that needs Doppler or an independent probe. The authors do not discuss this.

Verdict: Plausible, untested in the strict sense of a neutral test of what the slope represents; the measurement itself was repeated (Royds) but its interpretation is contested (Hull 1906). Split: (i) a slope exists and its numbers are of order 10^2 to 10^3 m/s: reproduced, treated as Established as an observation; (ii) it is the velocity of material vapour: Never properly tested, with one contrary Doppler result on record. The 6-microsecond vs 560 m/s mismatch is an unresolved internal inconsistency of about 50 percent in the abstract.

Claim 2: moving-film method records inclined/curved metal lines, air lines straight, so slope measures emitter motion

a. Consistent? The kinematics are right (see worked example). Air lines straight and slightly widened: the emitter is at rest and the discharge is brief, so each air line is a short-duration straight segment [calc/logic]. Concern: lines are described as curved everywhere, so the local slope changes along the line; "mean velocity" V1 is the average over 2 mm, V2 is "less rational" but "found to give consistent results" [read: p.2-3]. Averaging different quantities (V1 vs V2) across different rows of the results is a mixture of definitions, and the abstract does not say which one the Al/Bi/Cd/Hg numbers use [read; deep read]. Curvature: if the speed were constant the line would be straight; curved lines mean the speed changes (deceleration away from the pole, or, in Hull's reading, a front with varying propagation). The authors give no velocity-vs-distance profile; they decline to give a maximum velocity near the pole (the source of "do not allow us at present to give any opinion as to the maximum velocity" [read: p.2]). Also, finite slit width and image width convolve the line, so a line a few tenths of a mm in run is smeared; fine detail at 0.14 mm run (fast metals) is at the edge of what one can measure. b. Consistent with established experiments? The method is Dixon's explosive-wave method [read: p.1] and is standard in later streak spectroscopy; Kerr-cell time-resolved spectroscopy is described as following "since the time of Schuster and Hemsalech" [web, via Investigator, Springer Applied Spectroscopy]. Hull's contrary Doppler result remains. c. Tested? Repeated by Royds 1908 and 1910 using the same moving-film method [web, via Investigator]. No independent Doppler confirmation of motion on record in the Investigator's findings; the Hull Doppler test, if its geometry is right, did not confirm it. d. If true: nothing changes.

Verdict: Established as a recording technique (it reproduces and later became standard); the interpretation of slope as matter motion is Plausible, untested by an independent method, with Hull's contrary result.

Claim 3: discharge starts in air; the later oscillations pass through the metal vapour (against Feddersen 1862); evidence: a coil makes air lines nearly vanish

a. Consistent? The qualitative story is internally coherent and uses simple circuit physics: a series inductor L in a capacitor discharge lengthens the period, T = 2 pi sqrt(LC), and lowers the peak current, I_peak = V0 sqrt(C/L) (undamped), for fixed stored energy E = (1/2) C V0^2 [calc]. Worked example: C = 0.033 microfarad; L from 3 microhenry to 30 microhenry raises T from 2 to 6.3 microseconds and lowers I_peak by sqrt(10) = 3.2 times [calc]. The number 6 microseconds is comparable with the cadmium transit time, which makes the authors' argument (the longer discharge lets the vapour reach the gap) quantitatively sensible, though they do not do the arithmetic. Gap in the argument: the abstract has no numbers for the coil (inductance, jars, photographs), and "a great part of the energy of the current may then do useful work heating the metallic molecules" is a qualitative energy statement with no energy budget. Also, the first discharge cannot, mathematically, both "go through the air" and be absent: Feddersen's own oscillation analysis (discharge through a fixed resistance) is only modified, and no resistance model is offered for the changing vapour column. b. Consistent? Yes. The Investigator reports independent confirmation of the air-line suppression by self-induction: Schaeffer 1908 (ApJ 28, 121) credits Schuster and Hemsalech with finding that "sufficient self-induction in the discharge circuit causes the air lines to disappear" [web, via Investigator], and Hemsalech's 1901 thesis extends it to 14 metals in two groups (Fe, Mn, Co, Ni strengthened by self-induction; Cd, Zn, Mg, Al, Sb, Sn, Bi, Pb, Cu, Ag weakened) [web: https://archive.org/details/recherchesexpri01hemsgoog, via Investigator]. This is a second-hand source for a nuance the abstract does not contain: for some metals the lines weaken. It bears on how general the claim is. De Gramont's analytical spark spectroscopy is said to be built on the self-induction result [web, via Investigator]. The "air first, then metal" sequence is the same as the modern picture of breakdown followed by electrode-vapour plasma [memory: medium]. Qualification: the electrode question. Hemsalech 1906 reported vapour projected "solely from the positive electrode"; Schaeffer 1908 emphasised negative-pole disintegration; the 1899 abstract describes curves from both poles [web, via Investigator]. These three statements conflict as stated and the abstract's "both poles" is only seen in the no-prism pictures [read: p.6]. c. Tested? Yes: the coil experiment is described as "easily repeated" and was repeated by others (Schaeffer; Hemsalech; analysts). The Royds observation (calcium H and K lines starting at the gap centre and propagating toward the electrodes) [web, via Investigator, from the full paper] would conflict with simple "vapour leaves the poles" and should be kept in the record. d. If true: nothing; it is a refinement of Feddersen's account.

Verdict: Established for the self-induction effect on air lines (reproduced by other workers). The mechanism "later oscillations pass through the vapour" is Plausible, with partly independent support but not directly tested here; the energy statement is Not testable as stated (no budget).

Claim 4: vapour speed approximates the speed of sound, V = 80 sqrt(T/rho); cadmium 560 m/s, rho = 56, T about 2700 K; molecule not much lighter than normal

a. Mathematically? The arithmetic is correct: T = (V/80)^2 rho = 49 x 56 = 2744 K, printed 2700; and 80 sqrt(2700/56) = 555.5 m/s [calc]. The constant 80 is derived by me (the authors show none): c = sqrt(gamma R T / M) with M = rho x 2.016 g/mol = rho x 0.002016 kg/mol gives c = sqrt(gamma R / 0.002016) sqrt(T/rho) = 64.2 sqrt(gamma) sqrt(T/rho). For gamma = 5/3 (monatomic) this is 82.9; for gamma = 1.55 it is exactly 80 [calc]. So 80 is a rounded monatomic-gas value. Fine. Direct check at gamma = 5/3, M = 0.1124 kg/mol, T = 2700 K: c = sqrt(1.667 x 8.314 x 2700 / 0.1124) = 577 m/s [calc]. So the 560 m/s result is internally consistent with a monatomic cadmium vapour at about 2700 K. Mean thermal speed sqrt(8RT/(pi M)) = 713 m/s at the same T [calc], showing the "sound speed" and "molecular speed" are different quantities that the abstract uses as one word, "average molecular velocity". Hidden assumptions: (1) the observed slope speed equals the bulk sound speed; but a gas expanding into a vacuum (which the authors invoke: "flowing into a vacuum") has a front that can move faster than the sound speed; for a monatomic gas the free-expansion front can reach 2c/(gamma - 1) = 3c in the ideal limit [calc; memory: medium], though in practice the luminous part moves slower. They call the identification "approximate", and say "there is not much difference between these views", which understates a factor that could be 1 to 3. (2) The vapour is uniform in T; in fact the plasma is hot at the core, cooler at the edges. (3) Gamma "differences neglected." (4) The method makes T an output from the same V, so it is a hypothesis-dependent temperature, not an independent measurement; the authors themselves say only that T = 2700 "seems a possible value". It has no independent thermometer. Hemsalech's 1901 thesis argues spark vapour temperature is near arc temperature [web, via Investigator], which is the same order (arc about 4000-6000 K in air [memory: medium]); so the 2700 figure is at least not absurd. Because T depends on V squared, a 50 percent error in V (the 560 vs 830 m/s mismatch above) is a factor 2.2 in T: using 830 m/s gives T = (830/80)^2 x 56 = 6030 K [calc]. So the temperature is uncertain by about a factor of two from the paper's own numbers. The conclusion "the cadmium molecule is not much lighter than the mass determined near the boiling point" follows weakly: with rho taken as 56 (monatomic), the lightest cadmium species is already the atom, so the test can only exclude sub-atomic splitting; with rho = 112 (if Cd2) T would be 5490 K for the same V [calc]. The authors do not reverse the logic (use a temperature to find the mass); they assume the mass and get the temperature. With no independent T, V alone cannot give both. b. Consistent with experiments? Spark temperatures from later spectroscopy are of order 10^4 K or more at the core for fast sparks [memory: medium]; 2700 K is low. Not checked on the web; flagged for the physicist. c. Tested? Not tested by the authors; the Investigator reports only Hemsalech's argument that vapour temperature is near arc temperature. d. If true: nothing.

Verdict: Plausible, untested. The arithmetic is correct; the physical identification of measured speed with sound speed is a stated approximation, and the inferred temperature is not independent. Weak form (order-of-magnitude temperature of a few thousand kelvin): Plausible. Strong form (T = 2700 K and the "no dissociation" conclusion): Not testable as stated, because the mass and the temperature cannot both be derived from one speed.

Claim 5: no simple relation to atomic weight; lines of one metal differ; capacity effect unsettled

a. Consistent? The qualitative statements are consistent with the numbers given. Table I shows, for 6 jars at 0.51, 1.03, 1.54 cm: 416, 415, 435 (4925) and 529, 545, 492 (4811): "practically identical" [read: p.4]; I verify this (spread of 5 and 10 percent) [calc]. Capacity effect at 0.51 cm: 814 -> 556 -> 416, falling by a factor of 2.0 (4925) and 1.9 (4811) [calc]. A factor of 2 in V corresponds to a factor of 4 in T under the authors' own V proportional to sqrt(T), so the data imply temperature falling by about 4x with more jars, which the authors rightly call "very curious". Mathematically, one effect not discussed: with more jars the period T = 2 pi sqrt(LC) lengthens, so the film sees a longer discharge; line shape depends on how the discharge is spread in time. The abstract's own comment that "the periodic time was rather too small" [read: p.6] suggests the oscillation interferes with the vapour motion, which would be a systematic error not a physical trend. The 1.54 cm values (723, 1061, 435?; 1210, 1526, 492?) vary by a factor of 3.5 and have no error estimates; the authors say they draw no conclusion. The "no simple law connecting V to atomic weight" statement rests on prose descriptions (Al > 3x Zn, Cd about Zn, Bi has lines at 1420 m/s and others at zinc's speed, 3793 slower) with the Al "not laying any claim to accuracy". Quantitative check: if the velocities were thermal, V proportional to M^(-1/2), then Al (27) vs Zn (65) predicts a ratio of sqrt(65.4/27) = 1.56 [calc]; the reported "over three times" is twice that; Cd vs Zn: sqrt(65.4/112.4) = 0.76, reported "almost identical"; Bi (209): predicted 0.56 of zinc, reported 1420 m/s for some lines (about 2.8x zinc) [calc]. So a simple M^(-1/2) law fails, as stated. But at equal temperature V depends on sqrt(T/M), so they are not testing a "law of atomic weight" but T, which differs by metal (vaporisation energy, and ionisation) [calc/logic]; the abstract does not attempt this, and the full paper may. Different lines of one metal: the 4925 and 4811 lines differ by 10-67 percent, with the doublet (unresolved) systematically LOW. The authors' explanation (cross-wire set on the more refrangible edge adds spurious deflection) is consistent in direction with the observed deficit but is not quantified. If the doublet's two components have a splitting delta_lambda, the spurious extra slope is bounded by the apparent width; the abstract does not give it. They explicitly reserve judgment: "We reserve the decision of this point until we have been able to apply greater dispersion" [read: p.3-4]. b. Consistent? Royds's summary that lighter metals move faster [web, via Investigator] conflicts in tone with "no simple law"; both can hold if there is a trend with exceptions (Bi), which the full paper must settle. c. Tested? Repeated by Royds (1908/1910) with a velocity table by metal and line [web, via Investigator], not read by me. Replication of the exact capacity-dependence is not on record in the Investigator's findings. d. If true: nothing; different velocities for different lines of the same metal would suggest different emitters (ion vs atom) or differing excitation, which is mainstream (ionic lines vs neutral lines) [memory: medium].

Verdict: Never properly tested for the capacity and doublet anomalies, which the authors themselves treat as tentative; "no simple relation to atomic weight" is Plausible, untested (stated qualitatively, uses the unseen full paper's tables). Not dismissed without testing; the tentative items were left for follow-up.

Weighing the Investigator's findings (kept regardless of verdict)

• Full paper (Phil. Trans. A 193, 1900, 189-213, DOI 10.1098/rsta.1900.0006) holds all tables and photographs; not in the project. Any conclusion about Al, Mg, Bi, Cd, Hg rests on one sentence each in the abstract. [web, via Investigator]
• Royal Society referee reports: Crookes (13 Feb 1899): "The paper marks an important step in understanding the electric spark"; Abney recommended for Phil. Trans. [web: https://makingscience.royalsociety.org/items/rr_14_230 ; .../rr_14_231, via Investigator]. These are contemporary assessments by interested fellows of the same society, not independent tests.
• Hull 1906: Doppler limit, "appears to be due not so much to motion of luminous particles as to the propagation along those particles of a condition of luminosity". The strongest recorded challenge to claims 1, 2 and 4. Geometry not yet checked. Single source; evidence against stated here and not omitted.
• Royds 1908: same alternative raised from the same laboratory; calcium H and K starting at the gap centre and propagating outward. Evidence in tension with a pure electrode-origin picture.
• Electrode origin: Hemsalech 1906 (positive only), Schaeffer 1908 (cathode), 1899 (both poles). Three statements set side by side.
• Record continuity: the work was continued (Hemsalech 1901, Royds 1908-10, Strutt 1913, 2018-2022 plasma citations); it did not go silent, so no "absence of evidence" anomaly arises here.

Summary table