2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/99407The original Jeans dispersion relation and instability criterion are derived by a mathematically well-defined limiting procedure. The procedure highlights Jeans' physical reasoning and vindicates the (in)famous ``Jeans swindle.'' A second, independent procedure is stated which yields the same result.11 pages; An expanded version of this preprint appeared under the title: 'The "Jeans swindle." A true story -- mathematically speaking.' with the journal reference as given belowAstrophysicstext