VOL. LXXXVHI.] PHILOSOPHICAL TRANSACTIONS. 3Q1 



great while; so that, in order to determine how much the arm is drawn aside, it 

 is necessary to observe the extreme points of the vibrations, and thence to deter- 

 mine the point which it would rest at if its motion was destroyed, or the point of 

 rest, as I shall call it. To do this, I observe 3 successive extreme points of a 

 vibration, and take the mean between the 1st and 3d of these points, as the ex- 

 treme point of vibration in one direction, and then assume the mean between this 

 and the 2d extreme, as the point of rest; for, as the vibrations are continually 

 diminishing, it is evident that the mean between 2 extreme points will not give the 

 true point of rest. It may be thought more exact, to observe many extreme 

 points of vibration, so as to find the point of rest by difFerent sets of 3 extremes, 

 and to take the mean result; but it must be observed, that notwithstanding the 

 pains taken to prevent any disturbing force, the arm will seldom remain perfectly 

 at rest for an hour together; for which reason, it is best to determine the point 

 of rest, from observations made as soon after the motion of the weights as 

 possible. 



The next thing to be determined is the time of vibration, which is found in this 

 manner: I observe the 2 extreme points of a vibration, and also the times at 

 which the arm arrives at 2 given divisions between these extremes, taking care, as 

 well as I can guess, that these divisions shall be on difFerent sides of the middle 

 point, and not very far from it. I then compute the middle point of the vibra- 

 tion, and by proportion find the time at which the arm comes to this middle point. 

 I then, after a number of vibrations, repeat this operation, and divide the interval 

 of time, between the coming of the arm to these 2 middle points, by the number 

 of vibrations, which gives the time of 1 vibration. 



To judge of the propriety of this method, we must consider in what manner the 

 vibration is affected by the resistance of the air, and by the motion of the point of 

 rest. Let the arm, during the first vibration, move from d to b, fig. 3, and during 

 the 2d from b to d; bg? being less than db, on account of the resistance. Bisect 

 db in m, and Bd in m } and bisect Mm in n, and let x be any point in the vibration; 

 then if the resistance is proportional to the square of the velocity, the whole time 

 ©f a vibration is very little altered; but, if t is taken to the time of one vibration, 

 as the diameter of a circle to its semi-circumference, the time of moving from b to 



n exceeds -i- a vibration, by — nearly; and the time of moving from b to m falls 



short of 4- a vibration, by as much; and the time of moving from b to x, in the 



2d vibration, exceeds that of moving from x to b, in the first, by - — ■ -„ 



supposing ad to be bisected in $; so that, if a mean is taken, between the time of 

 the first arrival of the arm at x and its returning back to the same point, this mean 



will be earlier than the true time of its coming: to b, by — „. 



& ' J 8b» 2 */bx x x$ 



The effect of motion in the point of rest is, that when the 'arm is moving in the 

 same direction as the point of rest, the time of moving from one extreme point 

 of vibration to the other is increased, and it is diminished when they are moving 



