460 BELL SYSTEM TECHNICAL JOURNAL 



Derivations of Formulas for the Transition Points 



The above general formulas for the reversion of power series will imw 

 be applied in the derivation of the formulas (21) and (22) for Dn and 

 /^i, in the body of the paper; and also in the derivation o{ certain 

 other fdrnuikts, iml iiulu(k-d iIktc. 



To outline the dL-ri\ati(jii of the f(jrinula (21) for /)„, denote 

 (m— l);r/2 by (/„ and Dn-il„ by t„. so that (14-Al becomes 



{dn + T„) tan 7„=X. (19-A) 



Now replace tan r„ by its known power series expression, and divide 

 both sides of the resultini; equation by d„: thus (19-A) becomes 



j^=r„+|r„=+ ' r,.^+,.^r,.^+,4r„^+T^r„=+ .... (20-A) 

 a„ d„ 6 .ia„ lo loa„ 



This is of the form (15-A), and hence can be reverted b>- direct applica- 

 tion of (16-A); the result is (21). 



An alternative fornuila for />„ can lie obtained by starlini; from 

 Gregory's series, 



tan^z) , tan't' tan'i' , /.-.« i \ x 



i; = tan2' s— + — h ^+.... (20.1-A) 



3 5 I 



Application of this to (19-A) enables the left side of that equation to 

 be expressed as a power series in tan r„; and when the resulting 

 equation is reverted by means of (16-A) and then t„ replaced b\- 

 D„ — d„ the result is 



+(B-s^)a)'-(s-.J+i)(s)'+--- '-"' 



It has already been noii<! that (21) is not \alid for » = 1 and hence 

 does not include the foniiula (22) for Di. To obtain this formula 

 for Di, start willi the eqiialinii 



Di tan /),=X. (21-A) 



obtained by seltiiij,' /; = 1 in (14-.A). Then replace tan Pi b\- it> known 

 power series expansion, thus obtainini; tiie etiuation 



This is of the form (17-A), and hence can be reverted l>y direct a|i- 

 plication of (18-A); the result is (22). 



