TRACT 37.

OF GUNNER?.

241

56. For Example. With the l|lb ball of 2 inches dia-meter, and 2000 feet velocity ; that is or 1-125, d = 2,

p, and —T, = 37153 = /A + q z , and

1 1 mu'*

v 2000; also m = -00000757, n = ‘00175: then — =231, and f- — 1154q = ^ (37153— p z ) = 154yj therefore 7-522. Also

V — — 12-21, the tangent to the arc of 85° 19', the length

q 1 o 0

of which is 1-4S907 Then 7-522 + I'489 = 11-199 seconds,the time of ascending by this rule. And when the velocityv is infinite, then the tangent -—- becomes infinite also ; con-sequently its arc is a quadrant 1-570', and therefore 7'522 x1-57 = 11-81 seconds, is the time of ascent with an infinitevelocity.

57. Exam. 2. For any y other weight of ball, as supposethe 24lb = Wj the diameter being 5'6, or more nearly 5-546= d. Here then d 1 = 30-758, — = -78029, = 103078

7 d 2 nid 2

= />*+ q z , p — 115^ as before, q =; */{ 103078 — p 1 ) = 299‘4 ;_ V f 1 __ jo- 76. If the velocity v = 2000 as be-

fore, then the tangent —- — 6-293, the arc of which con-° 9

tains 80° 58', the length of which is T413; then 10-76 x1-413= 15-20 seconds, is the whole time of ascent when'projected with 2000 feet velocity. Also 10-76 x T570 =16-89 seconds, is the time when projected with an infinite

problem viii.

58. To determine the same as in Proli, v, taking into the ac-count the Decrease of Density in the Air, as the Ball ascendsin the Atmosphere.

In the preceding problems, relating to the height and timeof balls ascending in the atmosphere, the decrease of densityin the upper parts of it has been neglected, the whole heightascended by the ball being supposed in air of the same den-sity as at the earth’s surface. But it is well known that theVOL. Ilf. If