584 PHILOSOPHICAL TRANSACTIONS. [aNNO 1779- 



the thing may be shown much more easily in the following manner. Suppose 

 RB, AB, (fig. 9) to be 1 axes, about which every point in the plane abpr tends 

 to move with velocities as the respective distances from the axes ; let pq, per- 

 pendicular to AB, be to PR perpendicular to rb, as the angular velocity of p about 

 RB, to the angular velocity of the same point about ab ; and let the velocities be 

 in contrary directions : then, I say, every point in the plane will move with a 

 velocity proportional to its distance from the axis pb. First, it is evident, that 

 any point c in the axis rb will move round pb with a velocity proportional to its 

 distance cm : for the point c, lying in the axis rb, has no velocity round rb, and 

 CM is proportional to on. Draw pc parallel to ab, then any point d in that line 

 will move with a velocity proportional to dv, which is perpendicular to pb, for 

 the following reason : the velocity of d is equal to the difference of its 2 veloci- 

 ties round the respective axes rb, ab, or the difference of d's velocity round rb, 

 and p's velocity round ab, since all the points in pc move with the same velocity 

 round ab. Draw dt parallel to cr, then this difference will be proportional to 

 PT, because the velocities of p round rb, ab, are supposed equal to each other, 

 and PT is proportional to dv, and every point in the plane moves round pb with 

 a velocity proportional to its distance ; and the same thing may be shown when 

 any point is taken without the plane abcpr. 



^17. Because any point c in the axis rb moves with the same velocity round 

 PB as it does round ab, the angular velocities round the 2 axes ab, pb, will be 

 to each other inversely as their respective distances on, cm ; and because 

 ON : cm :: pb : pc, and pr : pq :: pc : cb, it follows, that pb, the diagonal of 

 the parallelogram pgbc, will represent the angular velocity of the revolving 

 plane, when bg, bc, are taken to each other as the angular velocities of the 

 same plane round those respective axes. 



■^ ]8. From this it clearly follows, that the reason given by Simpson, in his 

 miscellaneous tracts, pages 44 and 45, of the difference between his own solu- 

 tion and that of Newton in the Principia, cannot possibly be the true one. " It 

 appears further,' says he, " by perusing his 39th proposition, that he there as- 

 sumes it as a principle, that if a ring encompassing the earth at its equator, but 

 detached therefrom, was to tend or begin to move about its diameter with the 

 same acce'ierative force or angular celerity as that whereby the earth itself tends 

 to move about the same diameter through the action of the sun, that then the 

 motion of the nodes of the ring and of the equator would be exactly the same." 

 The principle is certainly implied in Newton's proof, and is capable of the most 

 rigid demonstration, art. 16, 17. 



^ 19. It will be asked then, where is the fault of Newton's reasoning } How 

 comes his conclusion to be too little by above one half? It is acknowledged on 

 all hands that there is an error in his 3d lemma; but then the correction of that 



