1895.] Theorems on the Attraction of Ellipsoids. 215 



giving rise to red dendritic patterns. The author considers these 

 differences to be analogous to the differences observed in the experi- 

 ments of Oliver Lodge upon the photo-electric loss of charge first 

 observed by Hertz. 



VII. "Theorems on the Attraction of Ellipsoids for certain 

 Laws of Force other than the Inverse Square." By 

 E. J. ROUTH, F.R.S. Received May 11, 1895. 



(Abstract.) 



The object of the author is to find finite expressions for the 

 potentials of an ellipsoidal shell, and of a solid ellipsoid when the 

 law of force is the inverse /c th power of the distance, K being positive 

 or negative. It is shown in the beginning of the paper that the two 

 cases in which * is an even integer and an odd integer require dif- 

 ferent treatment. 



After discussing some special cases, we come to the first general 

 theorem, Supposing that K is even and that the shell is a thin 

 homogeneous homoeoid, the potential is found to assume very differ- 

 ent forms according as K is greater or less than 3, so that the law of 

 the inverse square is just on one side of the boundary. When /c>3, 

 the potential can be completely integrated, and an expression is 

 found containing |(c 2) terms, and involving only the differentia- 

 tion of an integral rational function of xyz of K 4 dimensions. The 

 general form at an internal point is 



where P = 



E = l-ctf-ftf-tf 

 1 & 1 d? 1 d- 



A ~" + ' 



When *: is < 3 the potential takes the form of a single integral 





where t = (2 K). This reduces to the ordinary well-known form 

 when t = 0, i.e., when the law is the inverse square. 



Proceeding next to a thin heterogeneous homoaoid, the density 

 being 0(1^2) where is a function of i dimensions, different cases 



