The poor quality of Google translate seems to be entirely due to the poor quality of the OCR in the linked pdf file. I cleaned up the OCR and then the first two pages are translated as follows by Google, without any post-editing of the text (I only removed the display equations and added $ signs for the inline formulas). I would think the translation is clear, with the exception of the technical term "neck method", which we would read as "saddle point method".
I want to give some applications of a very fertile approximation method, called the neck method; it makes it possible to obtain approximate formulas for integrals relating to trigonometric or exponential functions; such integrals occur in the wave theories and in all the problems that are treated by means of the Fourier integrals. I will study the following types:
These various integrals occur in the theory of diffraction of light.
The integral of Airy gives the fringes near a caustic: $A (\nu)$ corresponds to the case of a practically unlimited opening; $A (\nu, r, s)$ in the case of a rectangular diaphragm ($^1$). Consider a wave produced by an optical system affected by aberrations. We assume a rectangular diaphragm; we can then, decomposing the wave into spindles, replace the wave by its equator $EE'$.
Let $C$ be a point of the caustic $CC'$ and $OC$ the radius tangent to the caustic in $C$.
The difference of a point $M$ of the wave at point $C$ is of the form
By asking
relative to a point $Q$ located on the normal to the caustic in $C$, the difference of the steps is easy to calculate, if one supposes $Q$ neighbor of the caustic and the small opening (that is to say $M$ neighbor of 0). We find
The phase difference is
The whole text is 54 pages, so this is just 4%, but it only took a few minutes, so I imagine this is entirely doable if there is sufficient interest.