27S ATTRACTION. 1 M MIi:i.T.. 



i by New ! .'!.. i. Pjrop, 7<>. 71), 'i 



with respect to the infernal particle was extended l>y N< 

 to the case of a shell l><>nndea by similar and similarly situated 

 spheroidal surfaces (Principia, l.ih. i. Prop. 91, Cor. ,'i). The 

 proposition is also true when tin- shell is ooundcd by similar 



-imilarly situated ellipsoidal surfaces, which we ]>r 

 to demonstrate in the method given by Newton for spin--. 

 surfaces. 



I'!."*. Jf a shell of v < tensity le bounded by two > 



soidal surfaces which are concentric, #/'/////'//. "//</ rirqii 

 situated, the resultant attraction on an internal particle vai( 



Let the attracted particle P be the vertex of an inlinit 

 series of right cones. Let NMPM'N' and nmPm'ri be two 



ating lines of one of these y 



cones, and suppose the curves in 

 the figure to represent the inter- 

 section of the surfaces of the shell 

 by a plane containing these gene- 

 rating lines. The curves will be 

 similar and similarly situated el- 

 B, and by a property of such 

 ellipses, 



^/\=^f'N' and mn = iu'/i'. 



By taking the angle of the e,,iu> small enough, each of tho 

 two portions of the shell which it intercepts will he ultimately 

 & frustum of a cone, and being of equal altitude and having a 

 common vertical angle, they will exercise equal 

 P. (S-e Art. 209.) Similar considerations hold with r< 

 to each of the infinite series of cones of which }> 5^ the v 

 and consequently the resultant attraction of the shell van 



This result being true, whatever be the thickness of the 

 shell, is true when the shell becomes indefinitely thin. 



216. In a somewhat similar way we may establish the 

 following proposition which is due to Poisson ; the result" at 

 tion of an indefinitely thin shell bounded by two ellip- 

 soidal surfaces which are concenti '<tr, and similarly 

 f cd on an external particle is in the direction f f/tc axis 

 C'f the enveloping cone ?///'// has its vertex at the given par- 

 ticle. (Crelles Journal, Vol. Xli. p. 141.) Denote the extenial 



