THE ROYAL ARTILLERY INSTITUTION. 
161 
where “ P ” is the statical measure and cc f” is the acceleration— i.e., 
the velocity (feet per second) generated in one second—and conse¬ 
quently “ mf ” is the momentum generated in one second. 
A retarding force, such as the resistance of the air, is a negative 
acceleration ; all the reasonings, therefore, concerning acceleration apply 
equally to retardation. 
If we know the' acceleration (or retardation) produced in a given 
projectile during a known time, and if we find it to be constant for 
equal times, we can at once find the force which produces it by (a) ; but 
in the case before us it is not constant, for it depends upon the velocity. 
The precise law connecting the resistance of the air with the velocity, 
has been hitherto, and may be still, considered unknown ; for Professor 
Bashforth has not determined any invariable law, but he has shown 
that for the average velocities of projectiles it is nearly as the cube of 
the velocity,* and he has calculated tables for correcting the error in 
this assumption. 
Since, then, the retardation is variable, differential equations of motion 
must be used, which may be determined from, and are analagous to, 
the ordinary well-known equations. 
The ordinary equations are 
for uniform velocity 
s '== vt, v — 
r»=A .-./=p 
for uniform acceleration J ^ _ vt _ a „ 
from rest | 8 ~ ~~ ^ 5 
U 2 = W- 
■0) 
When the velocity is variable, it may be considered constant for an 
indefinitely small time, during which an indefinitely small space is 
described. Hence v — -r ,. 
dt 
When the acceleration is variable, it may be considered constant for 
an indefinitely small time, during which the increase of velocity is 
dv 
indefinitely small. Hence f — . 
-n . ds 
But v = — 
dt 
<Ps 
dt*' 
dv 
Also , /= v ~ 
v 
dv 
dt 
Ts 
dt 
dv 
V Js' 
* For low velocities, as the 6th power; for high velocities, as the square. 
