About an Einstein equation This is a question about an historical theory of gravitation, studied by Einstein quite a bit before he settled on General Relativity. At that time, Einstein did not know that gravity was a consequence of curved space-time. He identified the variations of gravity with the variations of light speed in a gravitational field. 
In March 1912, Einstein postulated a first equation for static gravitational field, derived from the Poisson equation 
$$\Delta c = kc\rho  \tag{1}~,$$ where $c$ is light speed, $\rho$ is mass density and $\Delta$ is Laplacian.
Two weeks later, he modified this equation by adding a nonlinear term to satisfy energy-momentum conservation :
$$\Delta c = k\big(c\rho+\frac{1}{2kc} (\nabla c)^2\big)~. \tag{2}$$ 
Einstein's argument is the following:
The force per unit volume in terms of the mass density $\rho$ is $f_a$
$= \rho \nabla c$. Substituting for $\rho$ with $\frac{\Delta c}{kc}$ [equation (1)], we find $$f_a = \frac{\Delta c}{kc} \nabla c~.$$
This equation must be expressible as a total divergence (momentum conservation) otherwise the net force will not be zero (assuming $c$ is constant at infinity). 
Einstein says: 

"In a straightforward calculation, the equation (1) must be
  replaced by equation (2)."

I never found the straightforward calculation. That's something that's actually hard for me!
addendum 
The solution given and explained by @Gluoncito (see below) answers perfectly my question. However, it is likely that it is not the demonstration of Einstein for at least one reason : It is not a straightforward calculation. Historically, Abraham, a german physicist, was the first to generalize the Poisson equation by adding a term for the energy density of the gravitational field (coming from $E=mc^2$). He published a paper in january 1912 containing a static field equation with the term : 
$\frac{c^2}{\gamma}(\nabla c)^2 $ different but not far away from the Einstein term. After the publication of Einstein, Abraham claimed That Einstein copied his equation. I believe Einstein was at least inspired by Abraham. To what extent, I don't know.  
 A: There is a derivation of the equations above given by Giulini (may be more pedagogical?), You can look at it at :
http://ae100prg.mff.cuni.cz/presentations/Giulini_Domenico.pdf
As you will see he arrives at the same equation (2) assuming the "variable speed of light" is actually proportional to the gravitational potential, as I first assumed, (no need of variable speed of light). Please note that in general relativity the speed of light is constant in local charts, and that`s enough for the theory.
ok, as asked by the operators I copy the main parts of the demonstration:
the field equation for the gravitational field in Newtonian mechanics is:
$$\Delta \phi = 4πG \rho$$,
the Newtonian force per unit volume (mass density x acceleration) is:
$$f = −\rho \nabla \phi$$.
Now, the work done against gravity to assemble a piece of matter $\delta \rho$ (along an incremental change $\delta \xi$ along the flow) is:
$$\delta A=-\int \delta \vec{\xi}.\vec{f} =\int \phi \delta \rho$$,
a small change in the density of matter produce a change in the gravitational potential:
$$\Delta \delta \phi = 4πG \delta \rho$$
with this the work can be written:
$$\delta A=\int \phi \delta \rho=\delta ( \frac{-1}{8 \pi G} \int (\nabla \phi)^2)$$
where the equality is given integrating by parts the lhs.
Thus it is possible to find the energy density of the gravitational field as:
$$\epsilon=\frac{-1}{8 \pi G} (\nabla \phi)^2$$
Now, the important point is that any source of gravitational field must be compatible with the principle $E=m c^2$.
Thus, (I jump to eq. 9) the mass equivalence of the gravitational field is:
$$\delta M_g=\frac{1}{4 \pi G} \int \Delta \delta \phi$$ 
*
Newtonian gravity fail this principle since the rhs is zero in absence of matter. 
So Einstein added the energy of the gravitational field (the $\epsilon$ calculated before) as a source.
$$\Delta \phi=4 \pi G (\rho-\frac{1}{8 \pi G c^2} (\nabla \phi)^2)$$
Then computes the mass term (a complicated integral I`m a bit lost here, eq. 11), and redefines for consistency with the work $\delta A$ the field:
$$\phi \rightarrow \Phi=c^2 exp(\phi/c^2)$$
with this definition the equation becames:
$$\Delta \Phi=\frac{4 \pi G}{c^2} (\Phi \rho +\frac{c^2}{8 \pi G \Phi} (\nabla \Phi)^2)$$
The redistribution of the c-factors is due to the redefinition of $\phi$. The $\Phi$ multiplying $\rho$ cancels when you replace everything by $\Phi$ and gets the original eqs. * above.
Well, this was the derivation of Giulini, not very pedagogical because of eq. 11. If I understand it I tell you. It could be better to read the original Einstein paper but I have it only in German. I'm sure Einstein was clearer at the end...
