270 
PRINCIPLES OF GUNNERY. 
the sign on the right-hand side of the equation being negative, since 
c/) diminishes as t increases. 
Hence, dividing ( 1 ) by ( 2 ), 
du _ cv^' 
g ’ ... (3) 
or, since v = n sec <£, 
du cvfc , 
, — — sec 4 0 ; 
d(\> g • 
or ~ — - sec 4 (kd<p. 
ift g T r 
Integrating, 
/ u du c A 7 
= sec4 ^’ 
or = .*. (4) 
where P a = 3 / sec 4 = 3 tan a + tan 3 a. 
Suppose at the end of the arc OP' the horizontal velocity is q, and /? 
the inclination of the direction of motion to the horizontal line; then, 
substituting, 
l=l + £(P a _!>,).. (5) 
Tneveiocity Suppose u Q the velocity (horizontal) at the vertex of the trajectory, 
tex of the" II, then it follows from the above that 
trajectory. 
_L = i,£p. 
11 c B 
or — = — — - Pa .( 6 ) 
r u o 9 
Substituting this value of \ in equation (4), it follows that 
1 = _L - - P , \ 
m 3 » 0 3 g * « 0 3 \ g V 
on ^ 
Suppose —°- = y, ..( 7 ) 
then 1 = -^ (1 ■— yP $); 
u 6 n Q 6 Y 
orM =(i^i|. (8) 
