THE ELECTRIC IVAl'E-TILTER 29 



r = i2 sin-' ^. = i cos-' (l + ^) . (5) 



7 



y 2 



= 2 sinh-i jr = 2 tanh"' , = cosh 



2 -.2 



■'('+?) 



77r + 2 cosh-' |-. = /tt + cosh-' (- 1 - ^) 



(5a) 



- + 2iog[j+^'-i-r-], 



(5b) 



= i^ — sinh ^ 



(l+l)., (5c) 



-v - — 7'^ + — 7^ ^ 7^ + . . . , if I 7 I < 2, (5d) 



^ 24^ ^ 640^ 7168 ^ ^••■-"17! ^ ^, w^; 



2 cosh-i g + / 2 sin-' ^, (oe) 



where7 = « + .73,2, = ^|(|)V(l+|)V^((|)V(l-|); 

 = cosh"' A + i cos-' ^> (of) 



where 1 + i 7^ ^ .v + /'v, 2// = V (-v+l)- + y- + V(.v- l)-+y-, 



":!-K'+^=)" = K-4)"-(^)^' 



, /I 1 1 ^\ =^^ • 1 ( series ,„, 



= ^ hr 7 coth - r at mid \ . ^ (6) 

 \2 2 / / shunt. 



The formulas leave indeterminate the signs of 7, k, F, and K, and 

 also a term ^i^ttw in 7 and T. The signs are to be so chosen that 

 the real parts are positive, or become positive when positive re- 

 sistance is added to the system. The indeterminate =*= i27rn can 

 be made determinate only after knowing something of the internal 

 structure of the unit network of which the artificial line is composed; 

 the conditions to be met are — absence of phase differences when all 

 branches of the unit network A'^ of Fig. 1 are assumed to be pure 



