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Wikipedia's Sophie Germain; Work in elasticity; Subsequent attempts for the Prize says:

Germain had derived the correct differential equation (a special case of the Kirchhoff–Love equation),31 but her method did not predict experimental results with great accuracy, as she had relied on an incorrect equation from Euler,18 which led to incorrect boundary conditions.

18 Gray, Mary W. (2005). "Sophie Germain". In Bettye Anne Case; Anne M. Leggett (eds.). Complexities: Women in Mathematics. Princeton University Press. pp. 68–75. ISBN 0-691-11462-5.

31Ullmann, D. (2007). "Life and work of E.F.F. Chladni". European Physical Journal ST. 145 (1): 25–32. Bibcode:2007EPJST.145...25U. doi:10.1140/epjst/e2007-00145-4. S2CID 121813715.

Question: Upon which incorrect equation of Euler did Sophie Germain rely in her work that won a prix extraordinaire from the Paris Academy of Sciences?

This thorough answer to Did Sophie Germain find a flaw in Euler's equations for elastic vibrations? goes into some detail but states only that:

It could have been one of Euler's boundary conditions for the vibrating rod extended to plates. But, according to Hill, already in 1813 the Academy was impressed enough with Germain's predictions of nodal lines and frequency ratios in Chladni's experiments for square and rectangular plates, and she does not mention a boundary value issue in the 1816 memoir.

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    $\begingroup$ Likely related publication: L. Euler, "De motu vibratorio tympanorum", Novi comment. acad. sc. Petrop. 10 (1764) 1766, pp. 243-260 (Eneström Index E302). Title translates to "On the vibratory motion of drums". I wonder whether this is truly a case of an incorrect equation (I assume Euler did made mistakes at times, being human), or more a case of simplifying assumption used by Euler that does not hold for Chladni's experiments. If it is the latter, this could be very tough to find without reading Germain's and Euler's publications with great attention to detail. $\endgroup$
    – njuffa
    Commented Jan 22, 2023 at 6:26
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    $\begingroup$ Maybe also (or instead) the very next paper: L. Euler, "Tentamen de sono campanarum", Novi comment. acad. sc. Petrop. 10 (1764) 1766, pp. 261-181 (Eneström Index E303)? The title translates roughly to "{A test | an examination} of the sound of bells". Right now I do not see any other publications by Euler that seem immediately topically relevant. $\endgroup$
    – njuffa
    Commented Jan 22, 2023 at 6:34
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    $\begingroup$ In her work, Sophie Germain, Recherches sur la Théorie des Surfaces Élastiques, Paris: Huzard-Courcier 1821, Germain specifically references Euler's Tentamen de sono campanarum on pp. viii, 5, 29. She refers to a different publication of Euler's from 1779 on p. 56, but it is not immediately clear which one: " Nous y parviendrons à l’aide de l’analyse dont Euler a fourni le modèle dans le mémoire de 1779 ". Since this publication is about modelling, it might be the one with "the equation". $\endgroup$
    – njuffa
    Commented Jan 24, 2023 at 8:14
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    $\begingroup$ A scan of Tentamen de sono campanarum is available from the Euler Archive here. $\endgroup$
    – njuffa
    Commented Jan 24, 2023 at 8:20
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    $\begingroup$ On p. 29 of her publication, Germain identifies Euler's 1779 publication as Investigatio motuum quibus laminae .... This is L. Euler, "Investigatio motuum, quibus laminae et virgae elasticae contremiscunt," Acta acad. sc. Petrop. (1779) 1782, pp. 103-161 (Eneström Index E526). Scan available from the Euler Archive here. Title translates as "An investigation of the motions by which elastic plates and rods vibrate". This seems to be a prime candidate for the "wrong equation". The paper is too lengthy for me to process. $\endgroup$
    – njuffa
    Commented Jan 24, 2023 at 8:41

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