MECHANICS. 31 
and there being no angle of elevation, the descending branch of the para 
bola alone presents itself. Fig. 27 represents, by comparison with the pro 
jection in the plane, AF, the case where, to attain a greater range, AE, a 
projection to a lower level takes place with an angle of elevation, BAD; 
fig. 28 shows, in its left hand side, the diminution of range with a greater 
elevation; the right hand exhibits much the same case as in fig. 26. 
The preceding remarks are in all strictness to be taken with regard to 
projection in a vacuum, which, however, never occurs in practice. The 
resistance of the atmosphere, in which all bodies move which are projected 
from the earth, changes not only the path but the velocity of projectiles, 
and is very difficult to calculate accurately ; only very dense masses, as 
balls of lead, iron, &c., approach in their motions to the laws of projection 
in vacuo, and this indeed in proportion to their size. The range in air is 
5 to 10 times less than in vacuo. The greatest range is attained by a much 
smaller angle than 45° (in cannon even at 20°); the highest point of the 
path is nearer the end than the beginning; the descending part of the path 
is therefore much steeper than the ascending. 
d. Centrifugal Force. 
In the preceding remarks it was assumed that the directions of gravitation, 
in all points of the path of a projectile, were ‘parallel to each other. This 
is no longer the case, however, when we come to consider the motion of a 
body about an attracting point, as, for instance, in the motion of the earth 
or of any other planet about the sun. In such motion (central motion) two 
forces are to be imagined as operating: the centripetal force, which inces- 
santly solicits the moving body towards the attracting centre, and the tan- 
gential force, which, if the centripetal force were to cease its action, would 
impel it outwards in a straight line in the direction of the tangent. It 
depends upon the proportion between these two forces whether the body is 
to move in an ellipse or in some other curve. 
If a ball fastened to the end of a string be whirled around, the string 
experiences a tension which increases with the velocity of rotation. The 
cause of this tension is called the centrifugal force. It always acts 
wherever rotation takes place about an axis, and consequently in the 
rotation of the earth on its axis; at the equator it is greatest, as here the 
velocity of rotation is yreatest, and opposed to the force of gravitation; at 
the poles it is zero. In experiments upon the centrifugal force, the appa- 
ratus represented in jig. 29, pl. 17, may be employed, called a centrifugal 
machine. By means of the winch, d, the horizontal disk beneath it is rotated, 
this rotation being communicated by a string, e, to a second disk of smaller 
radius; this latter disk must turn the quicker as its radius is less. With it, 
and in the continuation of its axis, turns the vertical axis, c. Ifa thin ring 
of brass be fastened to the lower end of this axis, the upper curve capable 
of moving freely up and down the axis, this ring, if circular when at rest, 
will assume an elliptical shape when in motion, and the shape will deviate 
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