NavList:

Antoine, you wrote:
"Still I would be interested to know whether such Chavenant equations correctly tackle the extreme case when the parallax in Azimut is equal to +/- 180°"

First, Dave Walden mis-typed. He meant Chauvenet. If you have not already done so, you can download both volumes of Chauvenet's treatise on positional astronomy through a link in the index I maintain on the NavList web site here: https://navlist.net/navbooks2.html (look for Chauvenet vol. I in the index; lunars are in chapter VII). Almost certainly, you already have this text, and you have read this section more than once.

The correction you're talking about here is tiny one --tiny even by the standards of lunars. This is known in English as the correction for oblateness which here means the correction for the ellipsoidal form of the earth. It amounts to only 0.0-0.1 minutes of arc in most cases, occasionally as large as 0.2 minutes of arc in extreme cases. It has been standard fare in the lunars clearing web app on my web site here, http://www.reednavigation.com/lunars/, for a dozen years, and if you want to see its impact on any particular sight you can turn off the oblateness correction in the settings right at the start. As you'll discover, it's a very minor matter in most cases. As for the "extreme case" that you describe where the parallax in azimuth can jump 180°, this is merely a coordinate problem, and it can be avoided or dealt with in several different ways. Chauvenet's solution is to treat this as a correction to the parameters in the lunar distance clearing process, and that means it's just a very small quantity. There's no issue with jumping coordinate boundaries.

Frank Reed
ReedNavigation.com
Conanicut Island USA