PRINCIPLES OF GUNNERY. 
263 
If cji be the inclination of the direction of motion of the projectile to Theinciina 
the horizontal line in the plane of the trajectory, at any time, t, then projectile 6 
tan (j) 
= dy_ = 
dx 
dy 
dt 
dx 
dt 
to the liori 
zontal line 
in the plane 
vertical velocity at that time _ ^ jectol}™' 
horizontal velocity at that time 3 
or tan 4> = 
_ V sin a — gt _ (T — 2t) 
V cos 
T cot a 
If J3 be the angle of descent at time T, then t—T; then 
tan j8 — 
T-2T 
T cot a 
— tan a. 
i.e.j with an unresisted projectile the angle of descent is equal to the 
angle of departure. 
The above formula, (6) and (7), are very useful for the approxi- Approxi¬ 
mate calculation of the path of a projectile in the air for low-angle “oiofiow. 
trajectories, where the vertical velocity maybe considered as practically angletra- 
unaffected by the resistance of the air. 
For angles of elevation up to 5° the trajectory for a given range 
may be determined with considerable accuracy by the use of Tables I. 
and II. (Chap. Y., p. 55). 
The remaining velocity at the end of the range is found by Table I. 
from the equation 
(P 
S = S 0 — 
W 
or Sv = S v + — S, 
10 
on the supposition that the projectile is moving in a horizontal line 
unaffected by gravity. 
C N 
Thus, if 0 be the point of projection, and B the point of impact on 
the horizontal line through 0, then the projectile is supposed to move 
along OB, and the remaining velocity at B is found from the above 
equation by means of Table I. Then from equation, 
W 
and by means of Table II. T (the time of flight over OB) is found. 
