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.