I'ltACT 37.

OP GUNNEllY.

307

and with of p, or the whole weight of the charge, the ve-locity is between 6 and 7 thousand, which is probably thenearest of any.

182. Another consequence to he derived from our the-orem for the velocity of the ball, is the proper charge forany given gun, so as to communicate to the ball the greatestvelocity possible. It is not difficult to perceive that the ve-locity will not be increased b v increasing the charge beyonda certain degree; because, when the barrel is almost full ofpowder, the hall will he out of the piece before the chargehas time to give it the full velocity; and, on the other hand,when the charge is very small, it is too weak to give the balla sufficient impulse. So that, by increasing the charge gra-dually from the smallest, the velocity communicated to theball will be gradually increased, till it arrive at a certain de-gree; after which, the velocity will be gradually decreasedagain, as the charge is more increased, till the barrel is quitefilled with the powder. Hence it follows that, in every gun,there is a certain charge which will give the greatest velocityto the hall; and that by either increasing or diminishing thecharge, the motion of the ball will be diminished. Now it isevident that the knowledge of this charge is of great import-ance in artillery : for, by this means the artillerist is enabledto discharge the ball with the greatest velocity; and some-times to save much powder, by knowing that a greater chargewould not communicate so great a motion, but perhaps muchless. Therefore, to find the charge which will give thegreatest motion to the ball, we must make the formula amaximum which expresses the velocity, or its fluxion equalto nothing, considering the length a of the charge as the va-rying quantity j

183. If, for this purpose, we take here the first formula,

v — a/( ^ ’ x hyp. log. •—); then by squaring and can-

celling the constant factors, we obtain a X hyp. log. — amaximum, or the hyp. log of (~) a maximum. The fluxion

x 2