182 



on an m- 



The homogeneous shell (h y , 2 ) exercises no attraction 

 \ *W 



terior mass. 



The external equipotential surfaces of the homogeneous infinite- 

 simal shell ( h. zy i +dk J, are the surfaces (h, k) y in which h is arbi- 

 trary and k invariable*. 



The potential of the homogeneous infinitesimal shell ( A 2 , , 



upon an exterior point, is 



dh 



n 



and upon an interior point, is 



(In these expressions ^(h) is e w J , and h at the lower limit 

 in the first, is the parameter of the surface (h, k) which passes through 

 the attracted point. The density of the shell is supposed to be unity.) 



/ &"\ 

 The potential of the finite homogeneous shell ( A 2 , *, J (density 



= 1 ) upon an exterior point (, rj, ), is 



h "> dh , S> h * dh 



in this expression it has been assumed (for simplicity) that h^ is inde- 

 pendent of k. Also h" y h 1 are the values of h corresponding to k n y k' 9 

 when h and k vary subject to the relation f (, 77, , h y k)=Q ; and k y 

 in the last integral, is the function of h y , 17, determined by this 

 relation. 



The differential equations (2) are satisfied in the case of the ellip- 

 soid. For if we put its equation in the form 



^ 2 4- ^ A- * 2 k 



it is evident on inspection that 



^_L^ + ^= 

 dx* dy* dz* 



* tt is known that the last two propositions imply the first two (see Mr. 

 Cayley's " Note on the Theory of Attraction," Quarterly Journal of Mathematics, 

 vol. ii. p. 338) ; though this is not the order of proof in the present paper. 



