Oxford Academic (MNRAS archive): Whittaker 1902 and Stoney 1903

Visit: 2026-10-07, Investigator run (STEP I). Site: https://academic.oup.com (Monthly Notices of the Royal Astronomical Society back file, scans provided by NASA ADS). How reached: Google Whittaker 1902 "undulatory explanation of gravity" Monthly Notices, then Stoney "Examination of Mr. Whittaker's" "Undulatory Explanation of Gravity". Both articles are free to read. Chrome's PDF viewer showed page 1 only and the extension could not read PDF text, so the PDFs were read through the web-fetch tool, which kept a copy in the session's tool-results folder, and the pages were rendered to images (pdftoppm, 110 dpi) and read. Copies were not put in research/files/: downloading needs the owner's approval. Download URLs are below.

1. Whittaker 1902: his own first statement of the gravity idea (priority 1)

E. T. Whittaker, "On the General Solution of Laplace's Equation and the Equation of Wave Motions, and on an undulatory explanation of Gravity", MNRAS 62 (9), 617-620, Supplementary Number 1902, published 14 July 1902. DOI 10.1093/mnras/62.9.617. URL https://academic.oup.com/mnras/article/62/9/617/1015465 ; PDF https://academic.oup.com/mnras/article-pdf/62/9/617/3316471/mnras62-0617.pdf . The scan carries the code 1902MNRAS..62..617H. [web]

This is the short announcement that comes before the 1903 Mathematische Annalen paper in the project folder. Three sections: - §1, Laplace's equation (p.617): the general solution of del^2 V = 0 is V = integral from 0 to 2 pi of f(x cos v + y sin v + iz, v) dv, with f an arbitrary function of two arguments. [web: page image p.617] - §2, wave equation (p.619): the general solution of del^2 V = k^2 d^2V/dt^2 is a double integral of f(x sin v cos psi + y sin v sin psi + z cos v + kt, v, psi); by Fourier's double-integral theorem "any solution ... can be analysed into simple plane waves". [web: page image p.619] - §3, "An Undulatory Explanation of Gravity" (pp.619-620). [web: page images pp.619-620]

What §3 says (all [web: page images pp.619-620]): - "distributions of simple waves exist such that, although each individual wave is periodic with respect to the time, yet the total disturbance at any point does not vary with the time." - The example is the same as in 1903: a particle emits spherical waves; the band of wavelengths between 2 pi/mu and 2 pi/(mu + d mu) gives (2 d mu / pi mu) sin mu(Vt - r)/r, where V is the speed of the waves. Once the waves have reached r (Vt - r > 0) the sum is (2/pi r) times the integral of sin y / y dy from 0 to infinity, which is 1/r: "independent of the time and ... everywhere proportional to the gravitational potential of the particle." - The physical reading, p.620: the field of a gravitating body can be analysed "by a 'spectrum analysis' as it were, into an infinite number of constituent fields"; the whole does not vary with time, but each constituent is "of an undulatory character". - Longitudinal, in his words: "the varying vector is a gravitational force perpendicular to the wave-front", compared with light, "in which the varying vectors consist of electric and magnetic forces parallel to the wave-front". A force perpendicular to the wave front is a longitudinal wave. - Shared medium: "The waves are in other respects of the same type in the two cases, and it seems probable that an identical property of the medium ensures their transmission through space." - Speed: gravity "propagated with a finite velocity, which, however, need not be the same as that of light, and may be enormously greater." - Limits he sets himself: "this investigation does not explain the cause of gravity; all that is done is to show that the propagation across space of forces which vary as the inverse square of the distance does not require for its explanation any other property of the medium than one which has long been known and accepted."

How it bears on the claims: - Claim 4 (gravity as longitudinal waves in the medium of light). The 1902 note is the earliest statement found so far, a year before the Mathematische Annalen paper, and it is the one that names the longitudinal character outright: the force vector is "perpendicular to the wave-front". Several sentences of the 1903 §5 appear here word for word (the "identical property of the medium" and "enormously greater" sentences). - The hypothesis put to the review (Q07, gravity as a longitudinal standing wave). Whittaker never uses the word "standing". His sum is a set of outgoing waves from a source that has been emitting for a long time, and the time dependence cancels only inside the sphere r < Vt. The "standing wave" wording in later writers (Barrett 1993 as quoted by Titleman; Bearden; see 2026-10-07-qeios.md and 2026-10-07-google.md) is theirs, not his.

2. Stoney 1903: a contemporary physical critique (priority 2: a response by others; here it argues against)

G. Johnstone Stoney, M.A., Hon. Sc.D., F.R.S., "Examination of Mr. Whittaker's 'Undulatory Explanation of Gravity' from the Physical Standpoint", MNRAS 63 (8), 424-428, June 1903. URL https://academic.oup.com/mnras/article/63/8/424/1008995 ; PDF https://academic.oup.com/mnras/article-pdf/63/8/424/3275971/mnras63-0424.pdf . Scan code 1903MNRAS..63..424S. Stoney (1826-1911) is the physicist who named the "electron". [web: page images pp.424-428]

What Stoney says: - He praises the mathematics: the Laplace solution is "striking and most useful", and the wave-equation result is "a singularly elegant proof" of a "recently discovered theorem" and "a most welcome addition to the previously known proofs" (p.424). In a footnote he suggests it may extend to crystalline (anisotropic) media and refers to his own "method of Reversal" (British Association Report 1901, p.570). - His objection is physical (pp.425-427). An ordinary superposition (harmonics of a string, a spectrum of light) adds solutions of one equation, and each component is a real physical event. Whittaker's resolution is different: it picks out "one particular and very special solution" of the wave equation (2) to stand in for Laplace's equation (1). - He differentiates Whittaker's component dV = (2 d mu / pi mu) sin mu(kt - r)/r with respect to r to get the force (his eq. 4): dR = -(2 d mu / pi)[ (sin mu(kt - r)/mu)(1/r^2) + cos mu(kt - r)(1/r) ]. So each component force has two parts, one falling off as 1/r^2 and one as 1/r, both "flickering in intensity at each point of space" at frequency mu. Only "one definite set of coefficients" (amplitude proportional to 1/mu) makes them add to a steady 1/r^2 force (p.426). (The derivative checks out: d/dr of sin(mu(kt - r))/r = -sin/r^2 - mu cos/r.) - Mathematically (p.427), r appears both inside and outside the integral, and with these coefficients the r inside drops out of the result, which "produces the semblance of a resolution". To take it as physics one must assume that each particle's attraction is of the "extraordinary kind" in his §§7-8, and "the improbability of this assumption is so overwhelming that no physicist can feel himself justified in seriously arguing from the supposition that it represents the state of things which exists in nature, however much he may be impressed by the mathematical ingenuity of the investigation." - Postscript (p.428): at the meeting where the paper was read, someone suggested that solar-system motions not accounted for by Laplace's equation (Mercury is the case in view) show that the equation needs changing "for bodies in relative motion". Stoney answers that if so, the change cannot be Whittaker's substitution of the wave equation, and he proposes known causes instead: meteoric streams, gas escaping from planetary atmospheres, and an electric cause, stronger at Mercury.

How it bears on the claims: - This is the earliest recorded physical criticism of claim 4 found so far. It is one man's argument in the RAS's own journal. Whittaker's answer, if he gave one, has not been found. The JFM review of Stoney's paper (2026-10-07-zbmath.md) records it as a dissent "from the physical standpoint". - Stoney's 1/r component is a concrete point for the reviewers: each Whittaker "constituent field" has a force that falls off as 1/r, like radiation, and these cancel only in the sum. That bears directly on a "standing longitudinal wave" reading. If the constituents were physically real and the cancellation not exact (a source switched on recently, a moving source, a cut-off in the spectrum), the leftover would be time-varying 1/r forces. Stoney does not draw this consequence; the inference is the Investigator's. - The postscript ties the 1902-03 discussion to the Mercury perihelion problem, the same anomaly Tisserand's Weber-law calculation addressed (History Vol. 1, see claims.md claim 5 and A07).

3. Nature 68:144 (1903)

The OUP page for Stoney's paper links to Nature 68, 144 (11 June 1903), https://www.nature.com/articles/068144a0 . In Chrome this is the "Diary of Societies" listing, behind a login wall. It is probably only the meeting notice for Stoney's paper. Not read.