[ 22 ] .-f 



... vw Tx c +rt' Ev vw 



vaient to this = whole X •^^— X 77=^ = X E X ^r^^ X -=r- 



Ev TO r NT Ev 



Ta: c'+</- t, ^»T X Pr X Ta; i^T -r . • 1 1 



X — — = — X E X 7^-pr, •'^°^'^' " great circles be 



TC)_ r 1 C^' 



conceived drawn through P, O^, and P, E ; (by Sph Trig.) cos. P E 



{vT) X Rad. (TO) = COS. PO, (Tr) x cos. Q^E (T;^). There- 



fore a motive force at Q, equivalent to the motive, efficient, cen- 



• ,- , ^ rr- c'^+d' ^ Tr < Pr X Ta;' ^, - 



tnfugal force of E = x E X 7:^-- ; therefore 



r TO^« 



the fum of all thefe quantities = the motive force at O, equivalent 



to the fum of all the efficient centrifugal forces, or the centrifugal 



force of the ring. But it is eanly fliewn, that the fum of all thefe 



c^+d^ ,^ TrxPrxTO,' c'^d' ,p 



quantities = x ^ R x ;=r7=; -=^ = X 2 R 



^ r TQ; r 



c^r" X TO* <^^ I r, TT »i .• r . r^ 



^ . '~' ■ — X 2" R- Hence the motive force at O, 



^ c^ +d"xTO^ r -^ 



equivalent to the fum of all the efficacious centrifugal forces, is 



cd 

 expreffed by the fame quantity — x i R> as the force at Q, 



equivalent to the whole motive, efficacious force on the ring. 

 Q.E.D. 



, Mr. Simpson has pointed out the miftakes in the folutions of 

 this problem propofed by M. Silvabelle and Walmelley ; but neither 

 is his own calculation entirely faultlefs ; and his conclufion 

 appears to be correal, only becaufc the errors in the premifes com- 

 penfate each other. Thus he fuppofes, that the whole motive 



force, 



