380 PHILOSOPHICAL TRANSACTIONS, [aNNO 1778. 



has been lately made by Mr. Smeaton, and is described at length in p. 7 '2, of 

 this volume. 



It does not appear, that D. Bernoulli attempted to measure an) thing but the 

 time of a's descent through any particular space; Mr. Smeaton has given both 

 the times of q's descent, and the proportions of the velocities acquired, in a va- 

 riety of cases. By moving the weights he makes use of nearer to, or farther 

 from, the centre d, he alters the lengths of the levers at which the particles act, 

 without increase or diminution of their number: he does the same with the circle 

 or axis nmp, and consequently the lever md ; and in every case, from the known 

 character of that ingenious gentleman, we may presume that his numbers are 

 safely to be relied on. His conclusions may receive some illustration from the 

 preceding theory. 



From the proportion f •./:: ap -f ^'p '• op, it appears, that the force which ac- 

 celerates the motion of q, or in Mr. Smeaton's figure, the weight in the scale is 

 to the natural force of gravity, in a constant and invariable proportion, as long 

 as the quantities a, b, p, and p, remain the same; and therefore let a descend 

 ever so slowly, its motion will be uniformly accelerated throughout, and the spaces 

 through which it descends will be as the squares of the velocities acquired, and 

 the times will be as the velocities themselves; and this is agreeable to what Mr. 

 Smeaton found them in his 2d, 3d, 5th, 6th, Sth, and Qth experiments. 



The general expression for the force which accelerates the weight in the scale 



.^ fly+ p ^^^ ^jjj ^^ different according as the quantities a, p, or b, are altered: 



F y. ap . 



but is always easy to be determined as soon as those quantities are known. But 



it is impossible to determine the magnitude of the quantity b in the different cases, 

 unless we have given the precise dimensions of tlie whole machine, and the spe- 

 cific gravity of the wood made use of; and therefore I confess myself to have 

 been puzzled in endeavouring to reconcile the I stand 2d, and other experiments 

 with the theory; for though I could not doubt a moment, that the general ex- 

 pression for the force was rightly assigned, and would always be found consonant 

 to experience, yet I was extremely surprised to find, that when the quantity a in 

 the 'id experiment was made exactly one-half of what it was in the first, the time 

 of descending through the same space came out nearly double of what it was 

 before, and the velocity the same. Now this I knew could never happen unless 

 the force in the first case was to the force in the 2d as 4 to 1 ; for when the spaces 

 described are the same, the accelerating forces are always as the squares of the 

 velocities, or inversely as the squares of the times. This consideration led me 

 to inquire further into the ratio of those forces in the case described, in order to 

 discover, if possible, whether they came any thing neai' that ratio, which of ne- 

 cessity they ought to do. 



