PRINCIPLES OP GUNNERY, 
271 
From equation (2), 
dt _ v __ u sec 3 (f) \ 
— _ ^ sec 9 - g 
and if T represent the time over the arc OF, integrating we have 
T= r u sec 3 0^; 
Jp 9 
or, substituting the value of u in equation (8), 
_ u 0 r a sec % Qdcf) 
T _ Uo r a sec 
^*4 (f—yP# 
W 
Again, 
dx 
dt 
— u, 
.*. x == J* udt— — f U - 
3 sec 3 (j) 
d(j); 
or if X represent the horizontal distance traversed in the arc OF, 
X 
__ p°- se< 
(T- 
secr (j) 
Again, 
(J (1 — y-^) 3 
~ w tan 6 ; 
dt 
d(j ). 
,( 10 ) 
so that if X represent the vertical distance traversed in the arc OP', 
u q P a tan 0 sec 3 0 
-gM 
dcp. 
(u) 
Equation (9) gives the time of the projectile over the arc OF, and 
equations (10) and (11) give the co-ordinates of the projectile at that 
time. 
If the cubic law were correct for all velocities, and consequently the 
coefficient c constant, the range and time of flight of a projectile could 
each be correctly determined by this method in two operations—(1) by 
calculating over the arc from the point of projection to the vertex, 
(2) by calculating over the arc from the vertex to the point of impact. 
But as this coefficient is not constant for all velocities, it becomes 
necessary to calculate the arcs in smaller parts, and to take the mean 
value of the coefficient for these arcs, and thus to approximate very 
closely to the correct solution. 
The above is the solution which Bashforth has adopted in his 
valuable treatise on the “ Motion of Projectiles/^ in which he has 
* “A Mathematical Treatise on the Motion of Projectiles,” by F. Bashforth, e.d., published in 
1873 by Asher & Co. 
Bashforth’ 
method of 
solution. 
