149, 150, 151] 



TIDAL FORCES. 



409 



On account of the third power Z) 3 in the denominator, the effect of 

 the moon is much larger than that of the sun, in spite of its com- 

 paratively microscopic mass. 



151. Ellipsoidal Homoeoids. Newton's Theorem. If we 



transform Laplace's equation to elliptic coordinates and attempt to 

 apply the methods of 135 to the problem of finding the potential 

 of a homogeneous ellipsoid, we are at once confronted 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 homo- 

 geneous matter bounded by 

 two similar and similarly 

 placed ellipsoids exerts no 

 force on a point placed any- 

 where within the cavity. Such 

 a shell will be called an 

 ellipsoidal homceoid. 



Let P, Fig. 141, be the 

 attracted point inside. Since 



the attraction of a cone of solid angle do on a point of unit mass 

 at its vertex is 



Fig. 141. 



/dm Cr^dadr 

 r = / - 2 

 r* J r a 



we have for an element of the homoeoid the attraction 



d<D(BP-DP), 



do(AP-CP) 

 in the other, or in the direction PB, 



in one direction, and 



Draw a plane through ABO, and let ON be the chord of the elliptical 

 section conjugate to AB. Since the ellipsoids are similar and similarly 

 placed, the same diameter is conjugate to the chord CD in both. 

 But CD and AB being bisected in the same point, 



AC = BD, 



and the attraction of every part is counterbalanced by that of the 

 opposite part. 



