876 BELL SYSTEM TECHNICAL JOURNAL 



Ax 

 of 52-B (see Fig. 3) will occur in the vicinity of Xc -\- — . 52^ at this point 



may be determined as shown below, 

 where : 



V^^ + 2 y' 2 xL h 2{2n - 1) ^ ^"^ „4( 2(2/^ + 1) + J' 



dB 

 dx 



Again retaining only the first term of the error function and dropping all 

 higher order terms of Ax 



2n-l -1 



n, r n=oo 2n— 1 n=« 



TT L «=i 4(2w + 1) „=1 



nxr. 



„ri 2(2w + 1) 



(6) 



■ 2 n=» 2n-l ^ '^ 



_ Ax "^ nXc 

 ~ ~ 1^ hi 2n + r 



52jB proves to be negative and considerably larger than 8iB for the same 

 magnitude of interval. Therefore the computed B will always exceed the 

 true phase in the interval x,- to x^ + Ax except above a value of x very near 

 to Xc + Ax where the straight line approximation crosses the true phase 

 curve. When Xc = .35 and Ax = .02, 82B is found to be —.0005 degree 

 from (6) above, and for Xc = .91 and Ax — .005, 82B is also found to be 



— .0005 degree. The 82B errors are therefore found to be much more im- 

 portant than the 81B errors. 82B errors are not accumulative, however, 

 and therefore increments of Ax of the above order of magnitude prove to be 

 sufficiently small to give the accuracy being sought, namely ± .0005 

 degree in B. 



An evaluation of the 81B and 82B errors for values of Xc greater than .9 

 is difficult due to the slowness of convergence of the series giving these errors. 

 For values of x^ between .9 and unity, however, the frequency of known 

 values of /^determined from (1) above and available as check points is suffi- 

 cient to check the adequacy of intervals insofar as 81B errors are concerned. 

 Furthermore an analysis similar to that given above for the determination 

 of the 5i/> and 82B errors shows that an interpolation of the slopes computed 

 for construction of the tables in question, to give the intervening slopes 

 necessary to cut the increments of Ax in half will give check points at Xc + 



Ax Ax 



— frequencies, with a 81B error (Xc + -7^ is then the termination of a straight 



