110 Mr Fisher, Densities of the Earth's Crust 



Hence expanding, integrating, and taking the limits, the 

 vertical attraction at A can be expressed in a series of the form 



™ g /iW+(|)/*w +&•>•}*. 



which will be convergent when t is less than 2a sin - , i.e. when it 



is less than the chord of the sector. 



The limitation as regards convergency requires that the 

 distances from the frustra of the places at which we are com- 

 paring the vertical attraction of two similarly-shaped frustra 

 should be greater than the thickness of either of them. But since 

 the dimensions of continents and oceans are large when compared 

 with the depths which we propose to consider, this limitation will 

 not be important. 



Carrying the approximation to the (n — l)th powers of 



-) and (- 

 a) \a 



and eliminating a and transposing, we have the n — 1 equations 

 2S»0rf) -^(tV) =0, 



$ m (T?) -2 p (Tt'*) = 0, 



% m (rt n - 1 )-2 p (r't f ^) = 0. 



If we assume the sum of the densities to be equal to some 

 unknown number b, with that addition the equations may be 

 written in the general fo'rmf 



Tl +t 2 +t 3 +...+T n =b, x C n _!, 



Tjti + T2&2 T T"3^3 T • • • T T n t n = U, X C }1 _ 2 , 



TA 2 + T 2 t 2 2 + T 3 tj? +...+ T n t n * = 0, X C n _ 3 , 



Tj^- 1 + TA™- 1 + T^- 1 +...+ Tntr?- 1 = 0, X 1 . 



In fact, f-y (6) = 1 + sin - , 



/ 2 (0)=-/l + Jsin| + — 



4sin- 



* ,«v i 1 • i 5 



24 sm - 



f /m 1 ■ 6 X • ° . ■ 1 



64 sin 3 - 



<fec. =&c. Phil. Mag. Apr. 1894, p. 377. 



t See Todhunter's Theory of Equations, Art. 290. 



