TABLES OF PHASE 875 



Since A.r will be small compared to unity and since an error function is 

 being computed it is permissible to take only the 1st term of the difference 

 between the true phase and the computed phase, i.e. the Ax^ term, and drop 

 all higher order terms of Ax. 

 Then : 



81 B 



/r, ■i\ 2.n—l n^aa /n < \ ^n- 



n{2n — l)Xc A .3 \^ "(2''^ ~ ^)-^<= 



Ax' Z ^" ~ '^ : - A.r^ Z 



3{2n + 1) ■ ,tt 4(2w + 1) 



A.V -^ n\2n — \)Xc 



(4) 



E 



6x ;;rt 2w + i 



The equation (4) above for h^B gives only the error for a single increment 

 A.T of X = ///o. If the phase is known at x — Xa and x = .Xb and it is desired 

 to determine the phase at points between x = Xa and x — Xb then since 81B 

 always has the same sign the errors due to successive increments of x will be 

 cumulative and the total error at x = x & will be n times the average of the 

 diB errors of each increment of Ax between Xa and Xb where n is the total 

 number of equi-increments of x taken between Xa and Xb- However, since 

 the individual 81B errors decrease as the cube of Ax, the individual errors 

 will decrease as the cube of the number of increments taken between the 

 two frequencies at which the phase is known, whereas the cumulative 81B 

 error will increase only in proportion to the iirst power of n. Therefore, 

 the net result will be a vanishing of the cumulative error inversely as the 

 square of the number of frequency increments taken to approximate the 

 curve in the interval in question. It therefore follows that the accuracy 

 of the proposed method of building up the function, in so far as the phase 

 at the terminals of the straight line segments is concerned, is limited only 

 by the number of increments of frequency selected for the summation. 



In order to determine the actual magnitude of errors to be expected 81B 

 was computed for Xc = A and Ax — .02 and found to be only .000015 degree. 

 Since the total number of .02 intervals needed to be used between previously 

 computed values of 5 is 5, the total cumulative error in this region for 

 increments of this magnitude will not be greater than .0001 degree, which 

 is entirely satisfactory, since the accuracy being sought is ± .0005 degree in 

 B. For Xc = .9 and Ax = .005 the 81B error proves to be only .00001 degree 

 and since in this region the value of B has already been determined at .01 

 intervals by the more accurate series expansion technique referred to above, 

 only two increments are necessary between known values of B and therefore 

 the 81B error is sufficiently small. 



Having determined the order of magnitude of intervals necessary to keep 

 81B errors small, let us examine the errors due to the departure of the straight 

 line approximation from the true curve in the interval between Xc and Xc + 

 Ax. Since 81B will be very small it is anticipated that the maximum value 



