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The following communications were read : 



I. "On the Analytical Theory of the Attraction of Solids 

 bounded by surfaces of a Class including the Ellipsoid." 

 By W. F. DONKIN, Esq., M.A., F.R.S., F.R.A.S., Savilian 

 Professor of Astronomy in the University of Oxford. 

 Received September 2, 1859. 



(Abstract.) 



The surface of which the equation is 



f(*,y,*,M)=0, . . ...... (1) 



is called for convenience " the surface (h t k)" The space, or solid, 

 included between the surfaces (h lt k), (A 2 , k), is called " the shell 



\h' /' " an( * ^ at ^ nc ^ U( ^ e ^ between the surfaces (h, A- 2 ), (h, 2 ) is 

 called " the shell (h, ^ 2 Y " [This notation is borrowed, with a slight 



alteration, from Mr. Cayley.] It is assumed that the equation (1) 

 represents closed surfaces for all values of the parameters h, k, within 

 certain limits, and that (within these limits) the surface (h, k) is not 

 cut by either of the surfaces (h+dh, k), (h, k+dk). It is also sup- 

 posed that there exists a value h^ of A, for which the surface (h x , k) 

 extends to infinity in every direction. Lastly, it is supposed that if k 

 be considered a function of x, y, z, h, by virtue of ( 1 ), the two fol- 

 lowing partial differential equations are satisfied : 



dk 



in which q>(Ji) is any function of h (not involving k), and n is any 

 constant independent of h and k. The following propositions are 

 then demonstrated : 



The potential, on a given external point, of a homogeneous solid 

 bounded by the surface (h, k), varies as the mass of the solid, if h 

 vary while k remains constant. 



The potentials, on a given external point, of the homogeneous 



shells (h^ ^ 2 \ (X M are proportional to the masses of the shells. 



