162 



BELL SYSTEM TECHNICAL JOURNAL 



It thus is found that the value of Zm for all values of m greater than 2 can be 

 expressed in terms of Zq, Z\, and Z2. 

 Eq. (3.18) may be written in the form: 





z'^-dz 



Z3 = 



Zi 



The substitution 



V(Z - 2i) (S - Z2)(Z3 - z)(Zi - 2) 

 Zl = — (1 + ^o)Al , Zo = — 1 



/ (1 - /to)Ai , Case I) \ 

 \ 1, Case II / 



(1, Case I \ 



(1 - /feo)//fei, Casell, / 



Z2CZ3 — Zl) — Zi(Z3 — Z2)U^ 



Z = 



reduces the integral to 



^m — 



Z3 — 2i — (Z3 — Z2)m2 



— Zl) *'o / 



du 



h V(24 - Z2)(23 - zO h ^7 (73 -,2)(1 _ ^^2 -) 



where: 



tl = 



Z3 — Z2 

 23 — Zl 



2 (Z4 — Zi)(23 — Z2) 



X = 



(3.20) 



(3.21) 



(3.22) 



(3.23) 



(3.24) 



(3.25) 



(24 - Z2)(23 - 2i) 



Hence if A', E and 11 represent respectively complete elliptic integrals of 

 the first, second, and third kinds with modulus «, and in the case of third 

 kind with parameter —t], we have immediately: 



2K 



Zo = 

 Zl = 



z,= 



kl-\/(Zi — Z2)(23 — 2i) 



2 [2i K -\- {Z2- 2i) n] 



ki\/{Zi - 22) (Z3 - 2i^ 



^lV(24-l)(23-2i) [^^ ^' + '^'(^^ - ^^^" 



(3.26) 



(3.27) 



(3.28) 



