Did Sophie Germain find a flaw in Euler's equations for elastic vibrations?

Question

I am a playwright working on a play about Sophie Germain. When Sophie was competing for the prix extraordinaire to find effective formulas to describe the vibrations of elastic surfaces, she believed a combination of formulas from Lagrange and Euler would work. During her efforts, she discovered that an important equation of Euler's was incorrect. I've been trying to find out what that equation was. Does anyone know what it might be? Or perhaps where I might look to find that information?

Answer 1

There were two episodes with Germain and Euler's mistake in elasticity, but neither one of them is of Germain discovering Euler's mistake. The first one is from 1811 when she was starting to work on prix extraordinaire and corresponded with Legendre on Euler's paper on elastic rods. Euler wrote an incorrect solution for certain eigenmodes, but it was Legendre who spotted that in one of his letters to her. The second episode is of Germain using an incorrect equation from Euler to derive boundary conditions in the 1816 memoir Recherches sur la théorie des surfaces élastiques (published with corrections in 1821) that won the prize, which apparently led to a disagreement with Chladni's experiments.

A lot happened in between and after the two episodes, as described in detail in Hill's thesis Sophie Germain: a mathematical biography. The Academy extended the contest several times. It was dissatisfied with Germain's 1811 submissions because "the true equations of the movement were not established", they were also littered with technical mistakes. Lagrange derived the correct equation for vibrating thin plates based on Germain's hypothesis before passing away in 1813, but did not engage with her on the subject. Germain's second, 1813, submission, which was also deemed unsatisfactory.

The prize was finally awarded for the lengthy 1816 memoir, the first ever awarded to a woman, but could not attend because the Academy did not allow women at its meetings (other than members' wives). It was still dissatisfied with Germain's derivation because of technical flaws, not that it informed her of that. All of this took place against the background of the Academy members shunning Germain, and not communicating their responses to her work adequately; unethical behavior of Poisson, who took advantage of his position as a judge with access to Germain's 1813 submission, and presented her equation for thin plates with a technically correct derivation without crediting her; etc.

The first episode is described in Langton's talk The Reception of Euler’s Elasticity: Letters from Legendre to Sophie Germain at the 2007 workshop The Reception of the Work of Leonhard Euler (1707-1783). Euler was solving a special case of the partial differential equation for a vibrating elastic rod $$\sigma\frac{\partial^2 y}{\partial t^2}+B\frac{\partial^4 y}{\partial s^4}=0$$ by separating variables, and for a rod pinned at both ends and at the midpoint the equation for the eigenfrequencies he gets is $$\sin\frac12\omega\left(\tan\frac12\omega-\tanh\frac12\omega\right)=0.$$ Sophie asked Legendre, her patron, to clarify the derivation.

"In the course of solving his $8 × 8$ system, Euler has to divide by $\sin\frac12\omega$. Later, when he works out the solutions corresponding to $\sin\frac12\omega=0$ and finds that some of them have $\sin\frac12\omega$ in the denominator, he multiplies that factor out again. Legendre, in his first letter to Germain, asserts that this is an error on Euler’s part. Later, in his second letter, Legendre has understood better what Euler did, but he now discovers an actual error in Euler’s work. The solution that Euler writes down for the case $\sin\frac12\omega=0$ is incorrect."

Gray's note is behind a paywall, but the essence is reproduced almost verbatim in Wikipedia, with some details added. Gray herself is not specific as to which incorrect equation of Euler was at fault in 1816:

"A third competition was announced, to which Sophie, after consulting with the mathematician Poisson, submitted a memoir in her own name. This time she won the prize in spite of the fact that the equation was not rigorously demonstrated and that the agreement with experimental data was not too close because of her employment of an incorrect equation of Euler."

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.