TRACT 37.

OF GUNNERY.

243

X

. , , 1 ‘ 2 '/ 000 , 127000 n-x , . ,,

tural number-— XD = i,or ——- = d, the den-

1'27000

sity of the air at the altitude x, putting d = 1 the densityat the surface. Now put 127000ra or nearly 55000 = c ;then will be the density of the air at any general

height x.

61. But, in the 5th prob. it appears that av z denotes the re-sistance to the velocity v, or at the height x, for the densityof air the same as at the surface, which is too great in theratio of c 4- x to c — x ; therefore av 1 x -—- will be theresistance at the height x, to the velocity v, where a =•000026 J. To this adding w, the weight of the ball, givesav 1 x HiHi w ^ 01 t ^ >e whole resistance, both from the airand the ball’s mass: consequently — x -—- + — will de-note the accelerating force of the ball. Or, if we includethe small part — or I, within the factor -—-, which willmake no sensible difference in the result, but be a great dealsimpler in the process, then is av x - + * — ,/the accele-

rating force.

Consequently

vv

Qgfx = 2gx x -—-

° C + X

av 2 + w . ,

x -, and hence

C + X

w -

— X —-

% av 1 + xa

, or by division,

- * +

2 c

- X

C + X

w

32a

X

—vv

V* +

w

a

62. Now the fluent of the first side of this equation is evi-dently - x + 2c x h. 1. (c + x’); and the fluent of the latterside, the same as in prob; 5, is X h. 1. (v z -j- there-fore the general fluential equation is — x + 2c X h. 1. (c +x ) — X h. 1. (v z + But, when x = O, and v = vthe initial velocity, this becomes 0 + 2c x h. 1. c ~ x

h. 1. (v 1 + —); therefore by subtraction, the correct fluentsare _ * + 2c xh.l.^^x h. 1. when the