Attraction expressed ly an algelraic Quantity. 327 



Jh which the fluent, with respect to r, must be taken from 

 ir = c to r = such a function of S, as the nature ot the 

 curve' ABC, bounding the base of the cyhnder, may re- 

 quire ; and then the fluent, with respect to 5, must be taken 



from 9 = 0, to 6 = y; where tt = 3,1415, &c. 



Now, if R be the radius of the hemisphere, it is easily 

 perceived that ^^^_^ 



^^,.^6)^^KrZ7^n.--6-tR-/-c^OFl='/2i-ircos.i^:7^i 



whence ^. 



by a first integration of which we get 



F = l/"! (^B^^ll)? _I!i^£iii-^ I COS. fl. 9 ; 

 ^7 I (2RcosJ)4 (2Rcos.ej^ J 



„ F=— Rtt H r^ cos. -9. 9 (f3). 



«r, 1^-3 ^'^ 3y (211 COS. 9)^ 



Here the first term is the attraction of the whole hemi- 

 sphere ; and consequently the other term expresses the 

 action of that part of the hemisphere which remains after 

 taking away the portion cut out by the cyhnder ; so that 

 the solution of our chief problem wil be attained by mak- 

 incr this an algebraic quantity. But T shall hrst show the 

 usS of the preceding formula, by seeking the attraction of 

 such a portion as is intercepted by the cyhnder m Vivum . 

 celebrated problem, where the base ABCH is a c.rck 

 Let the radius of the base be any line p less than R, we 

 have, by the nature of the curve, r = 2p cos. 9 ; so that 

 the term to be integrated is 



_^_4 r((R-p)2 co^ ^„g fij ^ _ i./'(R^f) 

 y (2R COS. 9)i -^-^ 



i^f X 2 cos.^ 9. 9 = - -3- (R ^ ?) ^f^> ^^^^^"^^ 



sought is the difference of the actions of the hemisphere itself, 

 id ofanotiier hemisphere, whose radius is (R-p) ^-pT" j • 



Let us now go back to equation (^), and if we only con- 



X4 *'"'^' 



