"Gauss's theorem can be established as follows. consider an attracting particle of mass m at the point P, and let a cone of small solid angle ω be generated by radii through P. This cone cuts the surface at the points Q(adsbygoogle = window.adsbygoogle || []).push({}); _{1}, Q_{2}, ... taken in order from P; the parts of the surface cut off by the cone at these points are S_{1}, S_{2}, ... and the outward-drawn normals are denoted n_{1}, n_{2}, ...

Now if P lies outside Σ (region of masses, each at points Q_{1}, Q_{2}, ...), the cone will cut S an even number of times, and the signs will be plus and minus alternately, so that the total normal force across S_{1}, S_{2}, ... will be zero. On the other hand, if P lies inside Σ, the cone will cut S an odd number of times, the sign being minus and plus alternately, so that the total normal forces across S_{1}, S_{2}, ... will be -mω."

Source: Spherical harmonic T.M.Macrobert

Help with this text! I don't understand the second paragraph as to why the cone will cut S an odd number of times, the sign being minus and plus alternately when P lies inside Σ vs when P lies outside Σ. I know the cone is just a mathematical object but I can't follow the author thought process here. How did he use the cone to cut the surface at Q_{1}, Q_{2}, ... ? Why will the normal force across S due to particle at P be zero when P is outside the mass but not inside the mass?

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# I Gauss' law for gravity

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