zbMATH Open (Jahrbuch über die Fortschritte der Mathematik reviews)

Visit: 2026-10-07, Investigator run (STEP I). Site: https://zbmath.org . zbMATH carries the Jahrbuch über die Fortschritte der Mathematik (JFM, 1868-1942), the German review journal of its day, so it gives contemporary reviews. A Cloudflare check ("Performing security verification") cleared by itself after a few seconds, with no CAPTCHA. Queries: au:Whittaker, E* py:1902-1904 (showed only JFM 33.0390.01, Modern Analysis 1902); Whittaker gravity py:1902-1905; ti:"partial differential equations of mathematical physics" au:Whittaker; citing documents rf:2658160. How I found it: HathiTrust full-text search for "Whittaker's undulatory" (2026-10-07-hathitrust.md) listed the Jahrbuch v.34 (1903) as a hit.

JFM 34.0827.01: review of the 1903 Math. Ann. paper (priority 2: a contemporary account)

URL: https://zbmath.org/?q=ti%3A%22partial+differential+equations+of+mathematical+physics%22+au%3AWhittaker (document JFM 34.0827.01). Reviewer: Prof. Wangerin (Halle a. S.). Section: Mechanik, Potentialtheorie. [web] - The review restates the Laplace solution (1), the series V = 1/2 + 1/sqrt(x^2+y^2+z^2) + sum over n of the terms with z +/- 2n i pi, and the wave-equation solution (2), which it calls a form "different from the known Poisson solution". It adds that the general solution can be seen as built from single plane waves F(lambda,u,v) cos or sin {lambda(x sin u cos v + y sin u sin v + z cos u + t/k)}, gives the solution (3) of del^2 V + V = 0 and the generalised Bessel function J_{m,n}. [web] - The closing sentence, on gravity (my translation): "Finally the author argues that, on the basis of the above decomposition of the general solution of (a) into single waves, gravitation too can be regarded as arising from longitudinal waves that propagate with very great speed in a medium." The German reads "aus longitudinalen Wellen, die sich mit sehr großer Geschwindigkeit in einem Medium fortpflanzen". [web] - Links: DOI; EuDML https://eudml.org/doc/158101 (an open copy of Math. Ann. 57; Chrome could not read the page, see sites.md). zbMATH records "Cited in 2 Reviews" and "Cited in 26 Documents".

Bearing: an independent reviewer in 1903 read the paper's physical claim exactly as claims.md claim 4 states it: longitudinal waves, very great speed, in a medium. Nothing suggests the claim was misread at the time.

JFM 34.0839.01: review of Stoney's critique

G. Johnstone Stoney, "Examination of Mr. Whittaker's 'Undulatory explanation of gravity' from the physical standpoint", Monthly Notices 63, 424-428 (1903). Reviewer: Prof. Lampe (Berlin). [web] - My translation: "Against Whittaker's paper in Monthly Notices 62, 619, which tries to derive an undulatory explanation of gravity from partial differential equations, the author raises objections from the physical standpoint and gives the grounds for his different position." - The review gives the page of Whittaker's gravity section as 62, 619 (the article starts at p.617). See 2026-10-07-oup-mnras.md for Stoney's argument.

Documents citing JFM 34.0827.01 (rf:2658160): related work, priority 3

26 citing documents. The first 15 shown include [web: zbMATH list]: - R. S. Ward, "Twistors in 2+1 dimensions" (1989). - M. Dunajski, "Oriented straight lines and twistor correspondence" (2005). - R. Pantilie, "The Penrose transform in quaternionic geometry" (2017). - T. Adamo, L. Mason, A. Sharma, "Graviton scattering in self-dual radiative space-times" (2023). - D. Baumann et al., "A new twist on spinning (A)dS correlators" (2025). - J.-P. Nicolas, "Conformal scattering on the Schwarzschild metric" (2016). - P. A. Martin, on Herglotz wavefunctions (2018); Banjai et al., Trefftz methods for the wave equation (2017); Baird and Wehbe, biharmonic maps (2018); others in elasticity and integrable systems.

Bearing: in mainstream mathematical physics, Whittaker's 1903 integral is cited as an ancestor of twistor theory and the Penrose transform. These are integral formulas that build every solution of massless field equations (including linearised gravity) from arbitrary functions, the same move Whittaker made for the wave equation. The Google search in 2026-10-07-google.md confirms this from Hitchin (Celebratio Mathematica, on Atiyah), Mason (Oxford notes) and Thompson, Swearngin and Bouwmeester 2014. This is the "different words for the same idea" lineage: the mathematics of claims 1-2 is alive in gravity research, but as a representation of fields, with no medium and no longitudinal gravity.