216 Theorems on the Attraction of Ellipsoids. [June 13, 



are found to arise according as *: is greater or less than 3 and i 

 greater or less than K 2. If >3 and i<K2 the potential can be 

 completely integrated. A finite expression is given containing not 

 more than K 3 terms, and involving differential coefficients of an 

 integral rational function of xyz. For an internal point the general 

 form is 



Vl = .1 _ 2__S^ -- (A' + lA>- 4 - 7i OV 



K 1 L(/c 3) L(/c 4 /z)\E/ 



where A and A' are two differential operators, and 2 implies summa- 

 tion from h = to K 4. The potential is also found for an external 

 point. If i> K2 or /c<3 the potential contains a single integral 

 which reduces to a known form when we can put K 2. There are 

 two standard expressions, one of which is of the form 



v 2 Trfi f_ C abc du M / ax by 



, *- 



where M = ^ + ._ + . ...,* = . 



D is a differential operator and R a quadratic function of xyz, which 

 are explained in the paper. 



Examples are given throughout to illustrate the mode in which 

 these general formulae are to be used, and full references to all other 

 writings on the same subject as far as they are known to the author. 



Passing on to a solid ellipsoid the potentials at an internal point 

 for both a homogeneous and a heterogeneous ellipsoid are discussed 

 both when K > and < 3 and finite expressions are found, which reduce 

 to known forms when K = 2. When the point is external and the 

 strata are similar ellipsoids with any law of density, expressions for 

 the potential are found for the cases K = 4, 6, 8, and 10. These are 

 reduced to depend on a single integral. Except when K = 4, these 

 can be completely integrated when the solid is homogeneous and in 

 some other cases. 



When K is an odd integer, there is a division of cases according as 

 /c is < or > 2. In the first of these cases, finite expressions for the 

 potential of a thin homoeoid are found (1) when homogeneous, and 

 (2) when -heterogeneous. There are corresponding expressions for a 

 solid ellipsoid. As explained in the text, these results differ in form 

 rather than in substance from some already known, but they are 

 treated in a different manner. 



In the second case, when K > 2, the integrations become very long. 

 Finite expressions for the potential of a homogeneous homoeoid are, 

 however, found (1) when the force varies as the inverse cube, and (2) 



