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XII. 



ATTRACTION OF BODIES; OF SPHEEICAL AM) NON- 

 SPHERICAL SURFACES ANALYTICALLY TREATED. 



i. THE attraction exerted by spherical surfaces and by hollow 

 spheres is first to be considered. If P be a particle situated 

 anywhere within A B D C, and we 

 conceive two lines AD, BC, in- 

 finitely near each other drawn 

 through P to the surface, and if 

 these lines revolve round a P b, 

 which passes from the middle 

 points a and b, of the small arcs 

 DC, and AB, through P, there 

 will two opposite cones be de- 

 scribed ; and the attraction of the small " circles D C, A B 

 upon P, will be in the lines from each point of those 

 circles to P, of which lines C P, DP, are two from one 

 cirle, and A P, B P, two from the other circle. Now^, 

 this attraction of the circle C D is to that of the circle 

 A B, as the circle C D to the circle A B, or as C D 2 to 

 A B 8 (the diameters), and by similar triangles C D 8 : A B 

 : : P C 8 : P A 8 . But by hypothesis, the attraction of C D 

 is to that of A B as A P* : P C 2 ; therefore the attraction 

 of D C is to the opposite attraction of A B as A P 8 , to 

 P C 2 , and also as P C 8 to A P 8 , or as A P 2 x P C 2 to A P 8 

 X P C 8 , and therefore those attractions are equal ; and 

 being opposite they destroy one another. In like manner, 

 any^ particle of the spherical surface on one side of P, acting 

 in the direction of a P, is equal as well as opposite to the 

 attraction of another particle acting on the opposite side, and 

 so the whole action of every one particle is destroyed by 



