30 PHYSICS. 
adjusting properly the lower one, that the velocity attained amounts to 2, 
4, 6, 8 inches in a second; being thus uniform. 
The laws already developed serve for the vertical motion of a body ; new 
ones must be obtained when the motion takes place in vacuo, in a direction 
forming any angle with the horizon. Starting then from the point of view, 
that all material points of the same body receive an equal progressive 
motion, it will be possible to restrict our attention to the laws of a single 
point of a body. 
Suppose (pl. 17, fig. 24) A to be the starting point, and AC the direction 
in which the body is thrown, this would move with equable velocity in the 
direction AC, if unacted on by gravitation. This, however, incessantly 
solicits it in a vertical direction, downwards, so that after one second it 
would be about 16 feet; after two seconds, 4.16, or 64 feet; after three 
seconds, 9.16, or 144 feet lower down than if this gravitation did not act. 
Calling the initial velocity a, and the angle, CAB, which the original di- 
rection forms with the horizon, a, then the projected body under the simple 
influence of the initial force, would in ¢ seconds traverse the path, ¢. a. and 
have reached the height ¢.@.sin.a. The force of gravity diminishes this 
height by gé’, and the formula becomes 7. a@.sin.a—g?’. It is evident that 
after a time the ascent of the body will change into a descent, and will 
finally return to the same horizontal plane from which it started. This 
takes place when?. a. sin. a —gt?=0, or gt? =t.a.sin.a, or after = csr 
& 
asin. o 
“—~ seconds, the 
2g 
body will have reached the highest point of its path, whose height amounts 
seconds. In the middle of this interval of time, or after 
asin. 20 The line of projection is therefore a pure parabola. The 
4g 
rectilineal distance of the point where the body again reaches the horizontal 
plane, from the point where it started, or the distance of projection, is = 
to 
az sin. 2a. 
mT eo 
it is greatest when 2x = 90°, or «= 45°; that is, when the body 
is projected at half of a right angle to the horizon. 
The theory of projectiles comes most into play in artillery, where it is 
desirable to determine, not only the path of the projectile in the air, but also 
the variation of range of the guns with the variation of the angle of eleva- 
tion. It does not come within the province of this work to adduce to any 
extent the comprehensive calculations and investigations necessary to deter- 
mine these paths; a few examples only are given of the modes of ascer- 
taining the lengths and greatest ordinates of the parabola in different cases. 
Thus, pl. 17, fig. 25, shows how the parabola is determined when the axis 
of abscissas of the projectile line. AK, is horizontal, and the direction of dis- 
charge deviates from the perpendicular, AB, where then the greatest ordi- 
nate passes through the vertex, D, of the parabola. In fig. 26 the projec- 
tion takes place from a height to a depth, the gun standing at A; the great- 
est ordinate is EB; the line of abscissas, AB, being no longer horizontal, 
204 
