PRINCIPLES OP GUNNERY. 
269 
The motion op a projectile in the air will now be considered. 
It is-assumed that the resistance of the air acts along the tangent to the Motion of 
trajectory of the projectile. This assumption is no doubt very nearly SfptfjSv 
correct for flat trajectories, but it cannot be relied upon with certainty 
for high-angle trajectories. The motion will also be assumed to take 
place in the plane of the trajectory. This, without doubt, is not quite 
correct, as it is well known that the projectile performs small gyrations 
round the mean path of its trajectory. 
The same notation will.be used as for the trajectory of an unresisted 
projectile. 
v the velocity at any point of the trajectory, 
u // horizontal component of v, 
// inclination of the direction of motion to the horizontal line in 
the plane of the trajectory, 
t a time, 
x // horizontal distance from the point of projection, 
y n the vertical distance. 
Suppose 0 the point of projection, a the angle of departure, Oil the 
ascending path of the projectile, H The vertex of the trajectory, Ox the 
horizontal line in the plane, of the trajectory, Oy the vertical line, P the 
position of the projectile at any time t, u its horizontal velocity. 
It is proposed now to consider the motion of the projectile over a 
component arc of the trajectory OP ', a and J3 being the inclination of 
the direction of motion to the horizontal line Ox, at the beginning 
and end of the arc ,p and q being the horizontal velocities— i.e., the 
values of u —at those points. 
It has been seen that for certain ranges of velocity the law of re- Equations 
tardation may be correctly stated to vary as the cube of the velocity, of motio11 * 
or to be equal to cv z . 
The equation of motion obtained by resolution. in a horizontal 
direction is, then, 
Px du o 
w = dt = - cv 008 
The equation of motion obtained by resolution in the direction of the 
normal to the trajectory is 
d(b 
*i = -^ oos 
0,0 ..(>($) 
