In his article De la pression ou tension dans un corps solide [On the pressure or tension in a solid body], Cauchy introduces a theory that allows to define Cauchy stress tensor. It looks like he makes a mistake in his formula for the "surface element" in his surface integrals, unless I am missing something. Can someone check and confirm this, please? I didn't find any errata or comments about this apparent error.

The "surface element" in his surface integrals (in Cartesian coordinates) is written as $\cos\gamma\,dy\,dx$, where $\gamma$ is the angle between the normal vector to the surface and the positive direction of the $z$-axis. On page 62, formula (4), he seems to claim that the surface area is given by the integral $$ \int\!\!\!\int\cos\gamma\,dy\,dx = s. $$ Of course this is not true: the surface area is given by the integral $$ \int\!\!\!\int\frac{dy\,dx}{\cos\gamma} = s, $$ and the "surface element" is $\frac{dy\,dx}{\cos\gamma}$.

Since one should be free to use any symbol for the "surface element" as long as one does not try to interpret it, such an error would normally not lead to any other errors, so it seems plausible that it might pass unnoticed...


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