Euler E384: Research on the inequalities of Jupiter and Saturn | English translation | OMH v1.0

English translation of “Recherches sur les inégalités de Jupiter et de Saturne” by Leonhard Euler (Eneström E384), titled “Research on the inequalities of Jupiter and Saturn.” The publication-title and prize-title variants are distinguished. Prepared for Open Math Heritage with OpenAI-assisted translation, mathematical transcription and LaTeX typesetting. OMH v1.0 is a machine-assisted draft that has undergone a full, fresh AI-assisted technical audit of source coverage, translation, formulas, tables and figures where present, and rendered output. This is not a claim of human scholarly review or certification of the historical mathematics. Version 2 is reserved for a fully reviewed edition. Original publication: Recueil des pièces qui ont remporté les prix de l’Académie royale des sciences, vol. 7 (Paris: Panckoucke, 1769), preceding publication title leaf and pp. 1–84. The sole geometric figure is on the publication title leaf. The prize essay is for 1752; its prize title uses “irrégularités du mouvement,” while the publication title uses “inégalités.” Sources: https://scholarlycommons.pacific.edu/euler-works/384/; https://archive.org/details/recueildespdes07acad. The NHM/BHL repository describes the historical volume as public domain; no new translation license is asserted. The original historical text is public domain by age; no publication license is assigned here to the new translation or editorial notes.