Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 41 



To prove the truth of what has just been asserted, we will begin 

 with determining the repulsion exerted by the inner sphere itself, on 

 any point p exterior to it, and situate at the distance r from its centre 

 O. But by what Laplace has shown {Mec. Cel. Liv. ii. No. 12.) the 

 repulsion on an exterior point p, arising from a spherical shell of which 

 the radius is r', thickness dr and center is at O will be measured by 



I'Kr'dr'p d_ (r + r)^-" - (r-/)^-" 



1 — ti.S — n'dr' r ' 



the general term of which when expanded in an ascending series of 



r' 

 the powers of — is, 



+ ^"- 2.3.4.5 27Tl ^-.r-prfr, 



and the part of the required repulsion due thereto will, by substituting 

 for p its value before found, become 

 %F . (n-2 \ ,o ,,,^ -2 + wx«.» + l M+2*-3xw + l*-l 



Si' . fn-2 \ , , ,„,: 



2.3 . 4 2« + l 



,'2\ -1 »-2 



X 



It now remains to find the value of the definite integral herein con- 



tained. But when 11 -\ is expanded, and the integrations are 



effected, by known formulae, we obtain 



(14) M 1 - ^ I (*' - O ' r''^^'dr'=/o' 2^ -jj {b'-r") ' .r"'+'dr' 

 ^^ "a'' r^, ,3 X "^ 'f n 3\ 



, 2^ + 3 y 2^ + 3.2^ + 5 ¥ 



* 25 + 3 + w a' 2* + 3 + ».2« + 5 + wa' "*■ ^'^ 



^,^, ^ (2) ^ r "^ 2) {^s+l-^n){l-xy'' r^ 



_, , ^, ,.„ . _ „ , , „ 'dx 



, 1J2S + 1 



2° • / « 3> 



r (.+ - + -) (1-.^)- 



Vol. V. Pakt I. 



