104 l'HJLOSOPHICAL TRANSACTIONS. [ANNO 1781. 



And the velocity of the recoil (= v) answering to this length is that of 2.Q880 

 feet in a second: consequently v — u, or 2.988O — 1.1358 is equal to 1.8522 

 feet in a second. 



But as the velocities of recoil are known to be as the chords of the arcs through 

 which the barrel ascends, it is not necessary, in order to determine the velocity of 

 the bullet, to compute the velocities v and u ; but the quantity v — u, or the 

 difference of the velocities of the recoil when the given charge is fired with and 

 without a bullet, may be computed from the value of the difference of the chords, 

 by one operation. Thus the velocity answering to the chord g.05 is that of 

 1.8522 feet in a second, which is just equal to v — u, as was before found. 



The weight of the barrel, together with its carriage, was 47-j- pounds, to 

 which three quarters of a pound is to be added on account of the weight of the 

 rods by which it was suspended, which makes w = 48 pounds, or 336,000 grains, 

 and the weight of the bullet was 580 grains, b is therefore to w as 580 is to 

 336,000, that is, as 1 is to 579-31, very nearly; and v = X w is equal 



to (v — u) X 579.31. 



The value of v — u answering to the experiments before-mentioned was found 

 to be 1.8522; consequently the velocity of the bullets, = v, was 1.8522 X 

 579.31 = 1073 feet in a second, which is extremely near 1083 feet in a second, 

 the mean of the velocities, as they were determined by the pendulum. 



But the computation for determining the velocity of a bullet on these prin- 

 ciples may be rendered still more simple and easy in practice; for the velocities of 

 the recoil being as the chords measured on the ribbon, if 



c be put equal to the chord of the recoil when the piece is fired with powder 

 only, and 

 c = the chord when a bullet is discharged by the same charge, 



then c — c will be as v — u ; and consequently as ■ X w, which measures 



the velocity of the bullet, the ratio of w to b remaining the same. 



If therefore we suppose a case in which c — c is equal to 1 inch, and the velo- 

 city of the bullet be computed from that chord, the velocity in any other case, 

 when c — c is greater or less than 1 inch, will be found by multiplying the dif- 

 ference of the chords c and c by the velocity that answers to a difference of ] 

 inch. The length of the parallel rods by which the barrel was suspended being 

 64 inches, the velocity of the recoil answering to c — c = 1 inch measured on 

 the ribbon is 0.204655 parts of a foot in a second; and this is also, in this case 

 the value of v — u; the velocity of the bullet is therefore v = 0.204655 X 

 579.31 = 118.35 feet in a second. Consequently the velocity of the bullet, ex- 

 pressed in feet per second, may in all eases lie found by multiplying the difference 

 of the chords c and c, by 118.35, the weight of the barrel, the length of the 



