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adling upon the ring is counterafled by the centrifugal force 

 arifing from this compound motion, and half only is efficacious 

 in accelerating the plane of the annulus round its diameter. As 

 Mr. Landen has not exprefsly demonftrated this propofition, I 

 am perfuaded I fliall afford the mathematical reader much gra- 

 tification, by here laying before him the following very elegant 

 demonflration, communicated to me by the learned Mr. Brinkley, 

 Profcffor of Aftronomy in the Univerfity of Dublin. 



Prop. If a rigid ring nq N O, revolves with two motions 

 (fig. 3.), one in its own plane, and the other about the diameter 

 q •J' O^; and if a motive force, adling at the point Q_, be fuppofed 

 equivalent to the whole motive force adling upon the ring, 

 then half this force is efficacious in accelerating the motion of 

 the point O, (in a diredlion perpendicular to the plane of the 

 ring) and the other half is confiuned in counteradling the cen- 

 trifugal force, arifing from the motion of the particles of the 

 ring about a momentary axis P T /». 



In the great circle n h let a point h (fig. 3.) be taken inde- 

 finitely near to «, and in the ring a point r, fo that n h and Or 

 may reprefent the angular velocities about the diameter and 

 the centre of the ring. Let d and c reprefent thefe velocities, 

 and r the radius of the ring. Draw rs perpendicular to the 

 . plane of the ring, and meeting the great circle h Qj in j- ; 



then 



