288 I'HILOSOPHICAL TRANSACTIONS, [aNNO 1778. 



addition of b: and g is corrected by taking always g -\ ^ b, or ^ -| ~— b 



nearly for each successive value of ^; or g is corrected by adding always -^ b to 



the next preceding value of g: and lastly, h is corrected by taking for its new 



values successively ^'^J , , , or by adding always ^ bh, or Z; nearly, to 



the preceding value of h; so that the 3 corrections are made by adding always, 



/; to the value of p, 



X b to the value of g, 



-^^^ X b to the value of /(. 

 P 

 Before proceeding to the experiments, it may not be improper to take notice 



of 3 seeming causes of error, which have not been brought to account in our 

 theorem for determining the velocity of the shot: and to examine here whether 

 their effects can sensibly affect the conclusion. These are the penetration of 

 the ball into the wood of the pendulum, the resistance of the air to the back of 

 it, and the friction on the axis: by each of these 3 causes the motion of the pen- 

 dulum seems to be retarded. The principle on which our rule is founded, sup- 

 poses the momentum of the ball to be communicated to the pendulum in an in- 

 stant; but this is not accurately the case, because this force is communicated 

 during the small time in which the ball makes the penetration; but as this is 

 generally effected before the pendulum has moved -^ of an inch out of its ver- 

 tical position, and usually amounts to scarcely more than the 200lh part of a 

 second, its effect will be quite imperceptible, and therefore it may safely be neg- 

 lected in these experiments. As to the 2d retarding force, or the resistance of 

 the air to the back of the pendulum, it is manifest that it will be quite insensible, 

 when it is considered that its velocity is not more than 3 feet in a second, that 

 its surface is but about 20 inches square, and that its weight is 4 or 500 pounds. 

 Neither can the effect of the last cause, or the friction on the axis, ever amount 

 to a quantity considerable enough to be brought into account in these experi- 

 ments: for, besides that care was taken to render this friction as little as possible, 

 the effect of the small part which does remain, is nearly balanced by the effect 

 it has on the distance of the centre of oscillation; for as this centre was deter- 

 mined from the actual vibrations of the pendulum, the friction on the axis would 

 a little retard its motion, and cause its vibrations to be slower, and the conse- 

 quent distance of this centre to be greater; so that the other parts of our the- 

 orem being multiplied by V h, or the root of this distance, which is as the time 

 of a vibration, it is evident that the friction in the one case operates against that 

 in the other; and that the difference of the two is the real eiffcacious cause of 

 resistance, and whicii therefore is either equal to nothing, or very nearly so. 

 These general cau^c- of errtjr in the principles of the experiments are therefore 



