430 BELL SYSTEM TECHNICAL JOURNAL 



tions. In Appendix A a power series formula has been derived for 

 D„ in terms of \ = L/L' and Z?„_i„ = (k — l)ir/2; if, for brevity, the 

 somewhat cumbersome (though expressive) symbol D„_i„ is denoted 

 by d,„ this power series is 



+s-A-i)a)'-(S-3i+£)(s)'+--<-> 



valid for m = 2,3,4, . . . but not for w = l. For m = 1, so that D„ = Di, 

 it is shown in Appendix A that the appropriate formula is ' 



n -./x/',_A4-iA^_i^'_-^8ix^ + ,ii229^ \ r22^ 



1^1 VA\^i 6-^360 5040 604800"^ 119750400 /' ^ ' 



Since, by (16), p = Di/'VX, the series for p is the series in the paren- 

 thesis; see also (23-A) in Appendix A. Alternati\e series-formulas for 

 evaluating Di and D„ are derived in Appendix A — formulas (23-A) 

 and (23.1-A) for Di, and (20.2-A) for /)„. It may be observed that 

 D„-d„<\fd„, that Di<V\, and that 1-/)<X (5. 



The smaller X, the more convergent are these formulas. Formula 

 (22) is highly con\ergent, even when X is as large as unity or even 

 somewhat larger. The convergence of formula (21) depends very 

 much on d„ and hence on n: when n is large, (21) is satisfactorily con- 

 vergent even for fairly large \-alues of X; but when w is small, (21) is 

 satisfactorily convergent only for rather small values of X. 



As a supplement to or as an alternative to formulas (21) and (22) 

 there will now be given a widely applicable formula of successive 

 approximation for D„, valid for all the \alues of ;; -including » = 1 — 

 and suitable even for large values of X. With l\ — d„ (the /)-widlh 

 of the wth transmitting band) denoted by r,„ this fi)rinula ((ieri\c<l 

 by Newton's general method of approximation) is: 



,, Xt„'+X sin T„' cos T„'—d„ sin^ t„' 



X-|-sin- r„' 



(22.1) 



wherein t„' is some approximate known \ahie of t„, and t„" is a more 

 accurate approximate value yieldeil by the formula. t„", in turn, 

 is to be used in the formula to compute a still more accurate ap- 

 |)nixinialc \alin- r„" '; and so on, through as many cycles as may be 



' From the sequence of signs in this formula, namely — I h, the sign of the 



next term is not evident. A similar remark applies to formulas (23-A) and (23.1-A) 

 in Appendix A. 



