http://en.wikipedia.org/wiki/Shell_theorem#Converses_and_generalizations
My question is about the 'general force' of gravity derived on this article (assuming the external shell theorem, Newton's three laws and that gravitational forces are central apply no torque, as evidenced by Kepler's second law). I have been trying to understand the mathematical proof of this to no avail. My attempt involved plugging a general $F(s)$ instead of $\frac{GMm}{s^2}$ in the proof shown below and equating the total force to $F(s)$, but I don't think this yields anything very useful because I need to solve
$$\int_{r-R}^{R+r} (r^2-R^2 +s^2) f(s)\ ds = 4r^2Rf(r)$$
for the function $f(s)$ and I'm not sure if this can actually be done. (The variable values are the same as defined in the link below, although $R-r$ and $R+r$ may be swapped for any positive constants). If we obtain the correct solution however, $f(s)=\frac{1}{s^2} + ks$, where $k$ is any positive constant.
http://en.wikipedia.org/wiki/Shell_theorem#Outside_a_shell
I don't really follow the equation (4) given in the citation below either, which supposedly proves this fact: http://articles.adsabs.harvard.edu//full/1985Obs...105...42G/0000043.000.html
Any heuristic explanation or approach to this proof would be helpful