CHAPTER V. 



ATTRACTION OF ELLIPSOIDS. ENERGY. POLARIZED 

 DISTRIBUTIONS. 



107. Ellipsoidal Homoeoids. Newton's Theorem. If 



we transform Laplace's equation to elliptic coordinates and 

 attempt to apply the methods of 88 to the problem of finding 

 the potential of a homogeneous ellipsoid, we are at once con- 

 fronted with a difficulty. It is not evident, nor is it true, that 

 the potential is independent of two of the coordinates, and that 

 the equipotential surfaces are ellipsoids. 



The following theorem was proved geometrically by Newton. 

 A shell of homogeneous matter bounded by two similar and 

 similarly placed ellipsoids exerts no force on a point placed 

 anywhere within the cavity. Such a shell will be called an 

 ellipsoidal homoeoid. 



FIG. 49 a. 



Let P, Fig. 49 a, be the attracted point inside. Since the 

 attraction of a cone of solid angle da) on a point of unit mass at 



