VOL. LXXXVIII.] PHILOSOPHICAL TRANSACTIONS. 403 



attraction of the case on the balls exceed that of -fth of a spheric inch of water, 

 placed at the distance of 1 inch from the centre of the balls; and the attraction of 

 the leaden weight equals that of 10.6 spheric feet of water placed at 8.85 inches, 

 or of 234 spheric inches placed at 1 inch distance; so that the attraction of the 

 case on the balls can in no position of the arm exceed , / 7 , of that of the weight. 

 The compuration is given in the Appendix. 



It has been shown therefore, that the force required to draw the arm aside 1 

 division, is greater than it would be if the arm had no weight, in the ratio of 

 1.0353 to 1, and therefore is = j^r of the weight of the ball; also, the at- 

 traction of the weight and copper rod on the arm and both balls together, exceeds 

 the attraction of the weight on the nearest ball, in the ratio of I.OI99 to 1, and 

 therefore is = 873 q 000d °f tne weight of the ball ; consequently d is really equal 



818N* l.OlQQ N a . , r N* i ,i /• 



to T5553 x m§556i> or 1wiir> msteadof T5683T' as b y the former <=°mputa- 

 tion. It remains to be considered how much this is affected by the position of 

 the arm. 



Suppose the weights to be approached to the balls ; let w, fig. 7, be the centre 

 of one of the weights ; it the centre of the nearest ball at its mean position, as 

 when the arm is at 20 divisions ; let b be the point which it actually rests at ; and 

 A the point which it would rest at, if the weight was removed ; consequently ab is 

 the space by which it is drawn aside by means of the attraction ; and let m(3 be the 

 space by which it would be drawn aside, if the attraction on it was the same as when 

 it is at m. But the attraction at b is greater than at m, in the proportion of wm 2 : 

 wb 2 ; and therefore ab = m(3 x — r = m(3 X (1 H — — ) very nearly. 



Let now the weights be moved to the contrary near position, and let w be now 

 the centre of the nearest weight, and b the point of rest of the centre of the ball ; 

 then Ai = m(3 X 1 + M, andBi = m(3 X 1 + ^- + ~ = ^Hx (l+£), 

 so that the whole motion b6 is greater than it would be if the attraction on the ball 

 was the same in all places as it is at m, in the ratio of 1 4 to 1 ; and there- 



fore does not depend sensibly on the place of the arm, in either position of the 

 weights, but only on the quantity of its motion, by moving them. 



This variation in the attraction of the weight affects also the time of vibration ; 

 for suppose the weights to be approached to the balls, let w be the centre of the 

 nearest weight ; let b and a represent the same things as before ; and let x be the 

 centre of the ball, at any point of its vibration ; let ab represent the force with 

 which the ball, when placed at b, is drawn towards a by the stiffness of the wire ; 

 then, as b is the point of rest, the attraction of the weight on it will also equal ab ; 

 and when the ball is at x 9 the force with which it is drawn towards a, by the stiff- 

 ness of the wire, is as ax, and that with which it is drawn in the contrary direction, 



3 F2 



