beneath continents and oceans compared. 



107 



substitute a different frustrum for a frustrum of the supposed 

 shell, such that the normal attraction still remains uniform, it is 

 clear that the substituted frustrum must contribute the same 

 amount to the whole normal attraction at any given distance (0) 

 from it that the frustrum of the original shell did. So if we 

 substitute another differently constituted but similarly shaped 

 frustrum elsewhere, and the normal attraction continues uniform, 

 it must contribute the same amount of attraction at any given 

 distance (0) that the frustrum of the original uniform shell did. 



Hence these two frustra, though differing from one another, 

 have the same effect on the normal attraction as frustra of the 

 supposed uniform shell, and therefore the same as one another. 



Consider a sector of a shell of sectorial angle a whose apex is 

 A, and let it subtend at the centre an angle 0. 

 Let its thickness be t x and its density r u and a 

 the radius of the sphere. Let it be assumed, 

 as we shall hereafter prove, that under certain 

 conditions the vertical attraction of the sector 

 at A can be expanded in a converging series of 

 the form 



Let this be overlapped by another sector of 

 thickness t 2 and density t 2 . Then the vertical 

 attraction at A of the composite sector will be 



aa 



*!+'£>/.«"+- 



+ T 2 



£>\ 2 \ ) 



' 



and in like manner if there is any number m of overlapping 

 sectors, their vertical attraction at A will be expressed by 



aa ■J2 m Tf-j/ 1 (0) + t m r 



t\* 



/.(*) + Ac. 



Now suppose another composite sector with p layers £/, t 2 ', &c. 

 of densities t/, t 2 ', &c, and a remaining the same. Then its 

 vertical attraction at A will be 



aa. js, T ' Q f, (6) + S, r (Q '/, (0) + &c. 



If now we suppose that the vertical attractions at A of these 

 two composite sectors are equal we must have 



aa 



~>" T (a) fl {6) + &C - + SmT (a)' fr {6) + &C " 



^r Q y; (0) + &c + v' ($jfr m + &e.J , 



