874 



BELL SYSTEM TECHNICAL JOURNAL 



In order to evaluate the errors involved in such a procedure let us refer 

 to Fig. 3 where a segment of the desired phase function to be constructed is 

 qualitatively represented on a large scale. It is assumed that the phase at 

 Xc, B{xc), is known and that it is desired to determine the error diB in phase 

 computed for Xc + Ax when it is assumed that the phase curve is a straight 



line from B{xc) at Xc, to Xc + Ax having a slope, — ( ^"c + ^ ) , the slope 



dx 



of the true phase curve a,t x = Xc -r — ■ 

 Then : 



dE I Ax^ 



hB ^ B(xc + Ax) - B(xc) - ^ U'<= + y ) ^^ 



where; 



9 1 -v* T 



B(x,) = - X. + ^ + ?^ + 



5 



25 



Bixc + Ax) = - [{xc + Ax) + i{xl + 3x1 Ax + SxoAx^ + Ax^) 



TT 



+ ^(^c + Sxt Ax + lOxl A.T- + lOxl Ax^ + 5xc Ax* + Ax^) + 



B{xc + Ax) — B{Xc) = - 



, x;A.r , XcAx' 



Ax + -^ — + — 



3 3 



Ax , XcAx , 2XcAx , 2XcAx^ , avAx' 



^95 



+ 



+ 



Ax" 



5 +25 + 



A.v I] 



,tri 2w 



2n— 2 n=oo 2re— 1 



+ Ax 2^ 



</x 



,. + f,A. 



1 n^l 2W + 1 



' n(2n — l)x\ 

 Ax\ /Ax 



+ Ax^ XI 



•Vc + 2x, 



3 (In + 1) 



2 



+ 



+ ^ I -v: + 4x 



2 



TT 



4 . 



x. Ax 



. , XoAx XcAx^ A.x^ ..^ — 



Ax + + + — + + 



3 3 12 5 



2.V, Ax" 



yjJi'/' ^Jkvv 



A 4 .5 



^^c^^ XcAx , Ax 

 10 10 80 



] 



n=oo 2n— 2 



AxE "" 



„^ 2w - 1 ±? 2w + 1 



n=«; /r^ i \ 2n— 2 



3 y n{2n - l)x, 

 + ^-^ £l " 4(2;. + 1) + 



(3) 



] 



l['-+-Kf)--:(f)" -•(¥)■+(¥)>-) 



