INVERSE METHOD OF DEFINITE INTEGRALS. 



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SECTION VIII. 



On ike Resolution of Equations involvings Definite Integrals. 



(l) By the Decomposition of the Integrals into Elements. 



40. The utility of the method of decomposition consists principally 

 in the verifications it offers to results obtained by other analytical pro- 

 cesses, the difficulty in the eliminations which it requires. 



Pm-i-^ Pm+if Pa Put/ 



Suppose a cylindrical shell exerts no force on any point in its axis 

 AB, the law of force tending to each particle of the shell being given, 

 but the law of density of the shell unknown, then the application of 

 the method of decomposition is this : 



Divide the shell into « + 1 equal portions by planes perpendicular 

 to the axis PiQi, P2Q2, &c. 



Let the density throughout each portion be supposed uniform, and 

 let the successive densities be pa, pi, p->....pn- 



Let the total actions on the points of division Qj, Q2...Q„ be equated 

 to zero, which will give n equations, and another will be obtained by 

 considering the mass of the shell. 



From these n + 1 equations, let po, p^, p^, &cc. be determined in terms 

 of ». .. 



Finally, make n infinite. 

 Vol. V. Part III. 



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