Davis: Present Status of Integral Equations 2,- 
equations as easily as we formulate the same problem in a differential 
equation with boundary condition. 
Thus we might study the problem of kinematics in the spirit of 
Volterra and Fredholm as follows: 
Let us designate by u(t) the distance passed over by a moving 
particle after a lapse of t seconds. Dividing the time interval into 
n parts,; Ab, during each one of which the velocity, v(t), is approxi- 
mately constant, we shall have as an approximation to u(t), the sum, 
u n (t) = S v(b) Ah- 
i — 1 
As n increases indefinitely the sum may be replaced by the 
integral, 
u (t) = S dt ' 
A 
Supposing the particle to have an initial displacement u Q at 
t = t 0 we can replace (8) by the more general formula: 
u(t) = u 0 + J* v(t) dt. (9) 
A 
By a similar argument v(t) can be calculated in terms of a 
variable acceleration a (t), 
v(t) = v 0 + J a(t) dt. (10) 
t o 
Combining (9) and (10), we shall have 
u(t) = U G + v Q (t - t p ) + J (t - s) a (s) ds. (11) 
t o 
The application of equation (11) is easily shown in the problem 
of the simple pendulum. 
The displacement, u, of the bob, P, in terms 
of 6 is 
u = L 6 . 
The acceleration effective in the motion of 
the pendulum is 
a g sin 0 . 
Let us suppose an initial displacement of u o =L0, 
