142 on THE MAXIMUlvr OF MECHANIC POWERS, AND 



4 A r* the area of the lefs one ; and^ therefore 4 A R' — 

 4 A r^— the area of the annukis. Now 8 A R R is the 

 fiux4on of that area, and 8 A R3 r the fluxion of all 



the pd\ Hence zv - ^fi^HZTrr ^IZIlZl 



4AR^ — 4Ar* 2R^ — 2r* > 



which when r vanilhes, oi* the whole -becomes folid> 

 is equal R-v/Tas in the laft problem. 



CoR. The fe6tors S a and S A, being to each 

 other as the areas of their refpective circles,, and there- 

 fore as the fquares of the diameter of thefe circles ; 

 and if A in this cafe reprefent a fimilar feftor of the cir- 

 cle whofe radius is unity, the fame refult will be had 

 with refpe£t to the parts A«, and B^, as in the former 

 cafe, for the diftance of the centerofgyration from 



the center S,: will in this cafe be ^ — ^ — -. And when 



r vaniflies fo that the fe6tors are complete fectors of the 



larger circle, than wzrv'—^R Vk- 



Prob. 6. Let A B be a beam uniformly thick, 

 having its point of fufpenfion at any variable diftance 

 from A, as at S; and let the beam be made to vibrate 

 with any given angular velocity : it is required to de- 

 termine that power, which a£ting at the extremity B, 

 "would have the fame angular force as the whole mafs 

 collefted into, and atting at, the center of percuflion. 



Let the length A B be v, ASzta', A 



and SB^-z^ — ;c ;- and the diftance of 

 the center of percufTion from S equal 

 y : then by the general expreffion 



J'- forcfcnhetody ^^'^^v if inftcad of 

 taking all the Jrd^ in the vrhole 

 beam, or fuppoiing all the particles 

 colle6ted into the center of percuf- 

 fion, we conceive a power / acting 

 at the extremity B fuch as m-uUipiied 

 by the fquare of its diftance S B, 

 (v — x), its force fhall be equal to all 

 the p d"" In the whole beam : thqn will 



y = 



