kennelly. — equivalent circuits of composite lines. 35 



Single Line Freed at Distant End. 



If the line AB is freed at B, its resistance at A, measured to 

 ground, is 



R fA = z coth ohms. (4) 



In the D. C. case the hyperbolic angle 6 is a simple real quantity, z 

 is a simple numerical resistance, and coth is the hyperbolic cotangent 

 of 6, a real numeric, obtainable from tables of hyperbolic functions. 

 Consequently, E fA is a simple resistance in ohms. In the A. C. case, 

 however, z is an impedance, or vector resistance, 6 is also a vector quan- 

 tity, and the hyperbolic cotangent of this vector is not ordinarily ob- 

 tainable from any tables thus far published. It must be computed, say, 

 with the aid of formula (142). The product of z and this cotangent is, 

 therefore, a vector resistance, or impedance, R fA . Similarly, all the 

 remaining formulas of this paper may be regarded as applying either 

 to D. C. or to A. C. cases ; but the D. C. reasoning will be followed, for 

 simplicity of numerical check. 



At any point P (Figure 1) along the line, distant I' km. from B, its 

 hyperbolic angular distance from B will be 



8 = I' a hyps. (5) 



The potential at P is 



Up = u B cosh 8 volts, (6) 



where u B is the potential at the far free end B, defined by the condition 



u B = u A /Gosh volts ; (7) 



whence 



"' = ^^1 V0ltS ' (8) 



The curve of potential, or voltage to ground, plotted as ordinates 

 along the line AB is, therefore, a curve of hyp. cosines, or a cate- 

 nary. In the A. C. case the curve of vector lengths, or numerical 

 values, of potential, plotted as ordinates along AB, is a sinusoid 

 superposed upon a catenary. 



The current-strength at the point P is 



. sinh 8 . . 



' p = ^sTnh~0 amperes, (9) 



where i A is the current entering the line at A. The curve of current- 

 strength plotted as ordinates along AB is, therefore, in the D. C. case, 

 a curve of hyp. sines, or curve of catenary-slope. 



