times a positive constant. For p>4rx/gT - t + nT, the integrand 
is equal to cos 8-2 /4x EMS 8° ¢/4x times zero, which is zero. But 
since the integrand is zero beyond this value of Bp, the integral 
ean be evaluated by integrating cos 8-2 /4x Se Sien 8° 2/4x from zero 
to 4rx/gT - t + nT as in the fourth expression in equation (5.12). 
Then the transformation given by equation (5.14) yields the final 
expression in equation (5.12). The theory of integration also 
shows that if 4rx/gT - t + nT had been negative, the integrals 
would still be correct as written. 
The integral given in the last expression for equation (5.12) 
is the sum of two known integrals. They are the Fresnel Integrals 
which are tabulated, for example, by JahnkeHmde [1945]. For any 
particular value of x, t, n, and T, the upper limit of integration 
is some number, and the table gives the value of the integral. 
Solution 
Each term in equation (5.8) can be treated in the same manner 
as the first term was treated. The final form of the solution is 
then given by equation (5.15) where G(x,t,T,n) and H(x,t,T,n) are 
defined by equations (5.16) and (5.17). These three equations then 
are the solution, because they can be evaluated for all values of 
ty n, and T, and for all x greater than zero. 
In order to show that it is the solution of the problem, equa=- 
tion (5.15) must reduce to equation (5.1) as x approaches zero 
through positive values of x. If the upper limits of integration 
are plus or minus infinity the values of the integrals are Salta rate 
by equations (5.18) and (5.19). 
Consider the expression for G in equation (5.16). Pick any 
=, aes 
