  The objective of this paper is to present a modern and concise new derivation
for the explicit expression of the interior and exterior Newtonian potential
generated by homogeneous ellipsoidal domains in $\mathbb{R}^N$ (with $N
\geqslant 3$). The very short argument is essentially based on the application
of Reynolds transport theorem in connection with Green-Stokes integral
representation formula for smooth functions on bounded domains of
$\mathbb{R}^N$, which permits to reduce the N-dimensional problem to a
1-dimensional one. Due to its physical relevance, a separate section is devoted
to the the derivation of the demagnetizing factors of the general ellipsoid
which are one of the most fundamental quantities in ferromagnetism.
