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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 [Formula: see text] (with N ≥3). The very short argument is essentially based on the application of Reynold's transport theorem in connection with the Green-Stokes integral representation formula for smooth functions on bounded domains of [Formula: see text], which permits to reduce the N -dimensional problem to a one-dimensional one. Owing 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.