THE IH'll.DING-VP OF SINUSOIDAL CURRENTS 565 



2C may be retained without serious error, even when the line cuts off 

 at a frequency w, 2ir, provided the line is sufficiently long, and thg 

 fre<|ucncy w, 2ir not too close to the cut-off frequency oijlir. 



The formal solutions of the infinite integrals (23) and (24) can be 

 written down by virtue of the following known relations: 



- /'* '^ sin t'\ ■ cos (//.,X)^ = C(.v^) + 5(.v=) , (25) 



- / ^ sin /'\ • sin (/;;X)= = C{x-) - S{x'), (26) 



where C(.v-) and 5(.v-) are Fresnel's Integrals to argument .v', and 

 .v=/';2/i,. 



-f ^sin(/'X-(/»3X)']=-5+ / ^(v)(/v (27) 



f Jo f^ o Ja 



where .4(3') denotes Airey's Integral (see Watson, Theory of Bessel 

 Functions) and y = (2i'irf'^{l','h3). 

 By aid of the preceding. 



"= { ^ + 11^3 + ^^+ ■ • + ■ • f • '1 C(-v)= + 5(-') } ■ (28) 

 "= 1 ^ + 11^3 + ^;^,+ •• + •■[•{ C(.0-5(.=) ; , (29) 



where n = (h} '2//.). 



This is the appropriate form of solution when {hi/hi) is less than 

 unity. 



On the other hand when {h^'hi) is greater than unity, the appro- 

 pri.ite form of solution is 



While no thj)rough investigation has been made, it appears prob- 

 able that for all values of the ratio /»3 7(2, either (28). (29) or (30), (31) 

 will be convergent. However, in practice it is sufficient for present 



