108 Mr Fisher, Densities of the Earth's Crust 



and therefore, 



+ &c. = 0. 

 This will be satisfied to the order of approximation of 



-) and (- 

 a/ \a 



independently of the value of 9, if 



It follows that two equal sectors, whose densities and thick- 

 nesses satisfy these conditions, will to the degree of approximation 

 named produce the same vertical attraction upon particles at their 

 apices, independently of the value of 9, i.e. independently of the 

 length of the sector. If a shorter length of them has the same 

 property, so will their difference. Hence the remaining frustra 

 will have the like property. Now since a and 9 may have any 

 values, neither of them being involved in the above equations, it 

 appears that, subject to the above proviso, we may make the pair 

 of frustra of any shape we please and of any size, and place them 

 at the same distance 9 from A. Consequently two frustra similar 

 and equal to one another in form at the surface, the layers of which 

 fulfil the above conditions, in regard to densities and thicknesses, 

 will produce equal vertical attraction the one to the other at any 

 point on the surface of the sphere equidistant from each of them, 

 and will therefore be suitable to constitute parts of a spherical 

 shell whose attraction is uniform over the whole sphere. 



We will now shew that the vertical attraction can be expressed 

 in a converging series of the above form, provided that the thick - 



Q 



ness t of the shell is less than the chord, 2a sin s , of the sector. 



2 



The vertical attraction of the sector at A will be 



rt re fa r i s i n 0( a — r cos Q) 

 jj J, (r'-2ar cos e + a*)* d « dedr < 



= TO. 



\ a 



- cos 9 I sin 9 



1 — 2 - cos 9 + — 

 \ a a 2 



\d9dr. 



Let - = u, and let cos = a. 

 a 



Then dr = adu, and dfx, = — sin 9d9. 



