162 
MINUTES OF PROCEEDINGS OF 
The differential equations of motion are these 
ds 
v = — j 
dt 
n _ dv 
J ~~dt 
_ d*s 
dt 9 ’ 
_ do 
V ds J 
Xc) 
The four equations (c) correspond to the four equations ( h ), and form 
the basis of all dynamical problems. 
W© see them employed in Chap. II. to calculate the- well-known 
equation to the path of a projectile in vacuo. 
y — x tan a 
9 & 
2 F 2 cos 2 a 5 
which is the equation to a parabola. The only force acting on the pro¬ 
jectile, in that case, is the vertical force of gravity; therefore, putting 
d 2 s 
O* x and y successively for s in -j-% ”, we obtain the formulas em¬ 
ployed—viz., 
dP 
d 2 x _ d 2 y _ ,, 
dP ~°’ dt * 
dy 
Integrating , we obtain -=- and ~ ; which are, respectively, the hori- 
tIt Clt 
zontal and vertical velocities at any time. The constants are determined 
d'^C el'll 
by putting t — o, when and become V cos a and V sin a respec¬ 
tively—the horizontal and vertical components of the velocity of 
projection. 
Eliminating u t” the above equation to the parabola is at once 
obtained. 
The process for determining the trajectory when the resistance of the 
air is considered, is precisely similar. 
3. Differencing. 
(Chap. III. Arts. 51-53.) 
Before considering the experiments on which Mr. BashfortlTs calcu¬ 
lations are based, it will be necessary to understand thoroughly the 
processes of differencing and interpolating. 
Whenever a function of x (such as x n , log x, sin x, and other more or 
less complicated expressions) is calculated for several successive values 
* g is negative because gravity pulls in the contrary direction to that in which y is measured. 
