70 BELL SYSTEM TECHNICAL JOURNAL 



the exact formulas for these three quantities must depend, in any 

 specific case, on the corresponding specific form of the function F in 

 equations (I) and (II) of the Introduction. However, general ap- 

 proximate formulas can be obtained when, as usual, the ^'s in (II) are 

 small enough compared with the K's to enable the right side of (II) 

 to be represented by the first-order terms of a Taylor expansion, so 

 that h will be given by formula (III), as a good approximation. Since 

 //, when so given, is a linear function of the chance-variables ki, - • • kn, 

 the formulas of Section 3 (in Part I) are directly applicable by setting 

 a = there, and identifying Z, br, Zr there with h, Dr, K here, and 



hence z and Zr tjiere with h — h and kr — K here, respectively^ Thus 



it is not necessary to write down here the formulas for h, li^, \h\'^. 



When h is approximately "normal," the chance that the unknown 

 value h' of a random sample consisting of a single value of^ lies without 

 a circle of specified radius centered at the mean value h of h can be 

 found by application of the graphs presented and described in Sub- 

 section 1.3. 



6. IMPEDANCE-DEVIATION AND REFLECTION COEFFICIENT OF A 



LOADED CABLE DUE TO LOADING IRREGULARITIES 



AND TERMINAL IRREGULARITY 



As represented schematically by Fig. 13, the physical system con- 

 sidered in this problem consists of a periodically loaded cable whose 

 loading-coil impedances and loading-section admittances, and also the 



< 1 1 2 2 



n n n j^i rL...JT__rL^ 



Y, Y2 Yr Yn, Y^ 



Near End, ^ar End 



OR 



Initial End 



Z = Impedance of system: W = 1/Z = admittance of system. 



T = Admittance of terminal apparatus. 



Xr = Impedance of typical loading-coil no. r. 



Yr = Admittance of typical whole loading section, no. r. 



X,Y = NoMIN.\L V.A.LUES of Xr,Yr. 



Y/2 = Nominal value of Yo and Yn. 



Fig. 13 



terminal admittance {T), deviate randomly from their nominal values, 

 so that the deviations are complex chance-variables; however, the 

 nominal value of the terminal admittance is not here restricted to 

 equality with the iterative impedance of the loaded cable, since such 



