VOL. LXXXVIII.] PHILOSOPHICAL TRANSACTIONS. 401 



whole force required to accelerate the arm, is the same as that required to accelerate 

 398 grains placed at each of the points d and d. 



Each of the balls weighs 11 262 grains, and they are placed at the same distance 

 from the centre as d and d; therefore the force required to accelerate the balls and 

 arm together, is the same as if each ball weighed ll66o, and the arm had no 

 weight; and therefore, supposing the time of a vibration to be given, the force 

 required to draw the arm aside, is greater than if the arm had no weight, in the 

 proportion of 11660 to 11262, or of 1.0353 to 1. 



To find the attraction of the weights on the arm, through ddraw the vertical 

 plane dwb perpendicular to vd, and let w be the centre of the weight, which, 

 though not accurately in this plane, may without sensible error be considered as 

 placed in it, and let b be the centre of the ball; then wb is horizontal and = 8.85, 

 and db is vertical and =5.5; let wd = a, wb = b } and let ^, or 1 — x = z; then 

 the attraction of the weight on a particle of matter at x, in the direction duo, is to 

 its attraction on the same particle placed at b :: b 3 : (a 2 + * 2 ^ 2 )*> or 1S proportional 



to rrra, and the force of that attraction to move the arm, is proportional to 



(a 1 + zH z )^ r r 



A X ^T, Z ? 3 ; but the weight of the deal rod and wire at the point x, was before said 



(a* + z 2 l 1 )^ ° r 3 



to be ?!f ■ ~A. , = - — r~ per inch ; therefore, if dx flows, the fluxion of the 



.. 7 - w 1642 + 1848z . 6 s x(l-z) , n v 



power to move the arm is = Iz X ^ h , a> + ^a = z X (821 + g24z) 



924a 1 



b*z x (821 + 103z + ) 



b* x(l-z) _ Vk X (821 + 103z - 924z g ) __ ' lj_ 



X (a* + /V)£ ~~ (a* + /V)f. (a* + /V)f 



a 1 

 9246?z x ( — + z 1 ) ■ 



g w hich as -— OS is — b * * (895 + I032) 9246> * 



The fluent of this is 



8956'z 1036 5 . 1036' 924/> 3 . h + J (a* + fV) JtI _ - 



= av(a + /V) " M«h¥) + 7* - -~ lo S- ~a — ; and the force 



with which the attraction of the weight, on the nearest half of the deal rod and 

 wire, tends to move the arm, is proportional to this fluent generated while z flows 

 from to I , that is, to ] 28 grains. 



The force with which the attraction of the weight on the end-piece de tends to 



move the arm, is proportional to A^ X -, or 29 grains; and therefore the whole 

 power of the weight to move the arm, by means of its attraction on the nearest 

 part of it, is equal to its attraction on 157 grains placed at b } which is —3-, or 

 .0139 of its attraction on the ball. It must be observed, that the effect of the 

 attraction of the weight on the whole arm is rather less than this, as its attraction 

 on the farther half draws it the contrary way; but as the attraction on this is 

 small, in comparison of its attraction on the nearer half, it may be disregarded. 

 vol. xviii. 3 F 



