THE ELECTRIC WAVE-hlLTER 27 



stant and zero iterati\e impedance at the lower or upper end of each 

 pass band and stop band, respectively, is also optional. This com- 

 pletely determines the lattice artificial line. No additional condition, 

 other than iterati\e impedance asymmetry, can be realized by re- 

 placing the lattice network b>- an\- four terminal network. 



APPENDIX 



Formulas for the Artificial Line 



Formulas for the propagation constant and iterative impedance of 

 the generalized artificial line, expressed in a number of equivalent 

 forms, have already been given in my paper on Cisoidal Oscillations,^ 

 but it seems worth while to deduce the formulas anew here from the 

 free oscillations of the detached unit circuit of Fig. 6, so as to complete 

 the physical theory by deducing the comprehensive mathematical 

 formulas by the same method of procedure. 



Ladder Network Formulas 

 Notation: 

 Zi, Z2 = series impedance and shunt impedance of the section of 



Fig. 4, which is equivalent to the general network N of 



Fig. L 

 V = A -{- iB = propagation constant per section. 

 K\, A'o = iterative impedances at mid-series and mid-shunt. 

 T = « + '^ = V Zi/ Z2 = propagation constant for uniform distri- 

 bution of Zi and 1 Z2, per unit length. 

 ^ ~ V ZiZo = iterative impedance of this same uniform line. 



In Fig. 6, the current is indicated as / and the potentials at the 



ends of the section as V,Ve~^. In order that the free oscillation may 



be possible the total impedance of the circuit (Zi -^ Z' -\- Z") must 



vanish; this determines the iterative impedance A'o. In addition to 



this condition it is sufficient to make use of two other simple relations: 



the proportionality of the potential drops in the direction of the current 



across Z' and Z" to 7J and Z", since they carry the same current 



(this determines the propagation constant F) ; and the equality of 



«"Cisodial Oscillations," Trans. A. I. E. E., vol. 30, pp. 873-90), 1911. In the 

 lowest row of squares of Table I, the iterative impedances and propagation constant 

 of any network are given in fi\-e different wa\s, involving one-point and two-point 

 impedances, equivalent star impedances, equivalent delta impedances, equivalent 

 transformer impedances, or the determinant of the network. The only typo- 

 graphical errors in Table I appear to be the four which occur in the first, third and 

 fifth squares of this row: in tfie values for K, replace {S, — S^) by (5, — Sr) and 

 place a parenthesis before U, — Ur)\ in the first value of AV replace 5,^ by S^-; in 

 the last value for T^ add a minus sign so that it reads cosh"'. 



