which achieves some maximum value when t is negative at the 
point t = tir as graphed in figure 2. For t>ty an increase 
in t results in a decrease of the argument of the sine curve. 
The minus sign at the front of the expression can be put in- 
side by adding wm to the term in brackets. 
Since the original problem was, in a sense, an initial value 
problem, the main point of interest will be in the behavior of 
S57) for t>O>t,. For this reason, consider the point P,. 
The terms which are constant for constant x, can be ignored and 
equation (4.16) can be written. Then if t is increased by the 
amount , Bas the new constant value will be equal to the old 
constant value minus 27, and S(7 ) will have the same value as 
before. Equation (4.18) can then be obtained by subtracting 
equation (4.17) from equation (4.16). Equation (4.18) trans- 
forms easily into equation (4.19) with the use of equation (4.13). 
Finally equation (4.20) can be obtained if equation (4.19) is 
solved for aes and the reciprocal of the solution is taken. 
By an exactly similar procedure, T.* can be found from 
equation (4.16) and equation (4.21). ly? is given by eouation 
(Aho22))6 
The only difference between equation (4.22) and (4.20) is 
that the second term under the radical is negative in equation 
(4.22). Thus for certain values of t' near t' = - t. + t,,, the 
M 
value of the term under the radical is negative and T> dis 
imaginary. Such a value of t' (or t) is shown at the point P, 
* 
in figure 2. An increase in the value of t by T) results in a 
decrease in the value of the argument by 217, but there is no 
S55 = 
