ag.algebraic geometry - Differential equation of line tangent to caustics

This problem (or rather, statement that I cannot understand) has arisen in a paper I have been reading "Geometry of Integrable Billiards and Pencils of Quadrics" by Dragovic and Radnovic. I'd be most grateful for any explanations of it (it may be a simple fact, but I'm not sure).

Let $Omega subset mathbb{R}^d$ be a bounded domain such that its boundary $partial Omega$ lies in the union of several quadrics from the (confocal) family $mathcal{Q}_{lambda}: Q_{lambda}(x)=1$ where

$$Q_{lambda}(x)=sum_{i=1}^{d}frac{x_{i}^2}{a_{i}-lambda}.$$ Then in elliptic coordinates, $Omega$ is given by: $$beta_{1}'leqlambda_{1}leqbeta_{1}'', ldots, beta_{d}'leqlambda_{d}leqbeta_{d}''$$ where $a_{s+1}leq beta_{s}'leqbeta_{s}''leq a_{s}$ for $1leq s leq d-1$ and $- infty < beta_{d}'<beta_{d}''leq a_{d}.$

Define $P(x):= (a_1 -x)ldots(a_d -x)(alpha_{1} -x)ldots(alpha_{d} - x).$

Now, we consider a billiard system inside $Omega$ with caustics $mathcal{Q}_{alpha_1}, ldots, mathcal{Q}_{alpha_d-1}.$

Why does the system of equations:

$$sum_{s=1}^{d}frac{dlambda_s}{sqrt{P(lambda_s)}}=0, sum_{s=1}^{d}frac{lambda_{s}dlambda_{s}}{sqrt{P(lambda_s)}}=0, ldots, sum_{s=1}^{d}frac{lambda_{s}^{d-2}dlambda_{s}}{sqrt{P(lambda_s)}}=0,$$

(which are apparently due to Jacobi and Darboux - I'd appreciate a modern reference because the only version I can find is scanned page-by-page in German), where $sqrt{P(lambda_s)}$ is taken with the same sign in all expressions, represent a system of differential equations of a line tangent to all the caustics $mathcal{Q}_{alpha_1}, ldots, mathcal{Q}_{alpha_d-1}$? Moreover, why does:

$$sum_{s=1}^{d}frac{lambda_{s}^{d-1}dlambda_{s}}{sqrt{P(lambda_s)}}=2dl$$ where $dl$ is an element of ``the" line length?

I found similar looking equations on page 4 of another paper (by Buser and Silhol), but cannot understand them either.

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