TRACT 37.

OP GUNNERY.

249

force, and - ^ - r — = ,/the accelerating force ; eonseq. w or

2gfx = 2tf\* X -, and * = — x -, the correct

fluent of which, by the 8th form, is x = ~ x h. 1. ———the general value of the space x descended.

72. Here it appears that" the denominator w — cv % de-creases as v increases; consequently the whole value of x,the descent, increases with ®, till it becomes infinite, whenthe resistance cv- is - r the weight of the ball, and whenthe motion becomes uniform, as before remarked.

73. We may however easily assign the value of x a littlebefore the velocity becomes uniform, or before cv- becomes= w. Thus, when cv 1 = w, then v = 252, as found in thebeginning of this problem. Assume therefore v a little lessthan that greatest velocity, as for instance 2*6 : then this va-lue of v substituted in the general formula for x above de-duced, gives t = 2927 feet, a little before the motion becomesuniform, or when the velocity has arrived at 246, its maximumbeing 252.

74. In like manner is the space to be computed that willbe due to any other velocity, less than the greatest or termi-nal velocity. On the contrary, to find the velocity due toany proposed space x, from the formula £=-^ x h. I. - w £ ua ■

Here x is given, to find v. First then — b. 1. take

therefore the number to the hyp. log. of which numbercall n ; then n := —-—; conseq. n» — n cv- = w, and Nat

— w = np® 1 . and v = V -a\ a general theorem for thevalue of v due to any space x. Suppose, tor instance, x is1000. Now 4g being = 64, re = it, and c = "0000176;therefore = P0012, and the natural number belongingto this, considered as an hyp. W. is 2*72162 = n; hence then

N — 1 ~ 1 °

v — V ~^r w — 201, is tiie velocity due to the space 1000,°r when the ball has descended 1000 feet.