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THEORY" AND PRACTICE
TRACT 37-
and to urge it, with its whole force and velocity; whereas infact it only acts with its relative velocity, or by the excessabove the velocity of the ball. On all these accounts then,it is rganifest that the velocity in the experiment must begreatly diminished, without receiving a previous theoreticalcalculation, to estimate its effect?.
167. Lastly, in the calculation it is supposed that thewhole strength of the powder is exerted in urging the ballforward ; whereas some part of the force must be employedin giving motion to the gross particles of the powder, andeven tffe geperated fluid itself; which consequently ought tobe deducted from that which was supposed to act on the ball,or its weight, either in whole or part, added to the ball, bywhich means the motion of this will be still further diminished.On all these, and perhaps other accounts also, it is manifestthat the experimented velocities of balls must be far belowwhat they ought to be, if the whole force of the inflamedpowder was employed and spent only upon the ball in givingit motion.
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168. To renew the calculation of the velocity then, so asto include in it, as well as that of the ball, both the weightof the charge of powder, and the flannel bags in which itwas contained. Now, because the elastic fluid of the in-flamed powder occupies, at all times, the part of the gunbore behind the ball, the middle of it, or nearly its centre ofgravity, will be moved with only half the velocity that theball moves with; and this will require the same force as halfthe weight of the powder, moved with the whole velocity ofthe ball. Therefore, in the result before derived, we arenow, instead of the weight of the ball w, to substitute thequantity p + V>, and then, instead of the first theorem, it
will now become v — v \ ~ v+v , - * hyp. log. —) for fhe ve-locity of the ball, p denoting half the weight of the powderand bag.
But when tlje given numbers are substituted for the lettersg, in, c, and the hyperbolic logarithms are reduced to the