MATHEMATICAL APPENDICES. 



205 



we get 



. _ e(A - kS) 



X -kY 



. . . . (8) 



~ XA + YB + k(XB - YA) 



Therefore the equivalent resistance, R, and equivalent react- 

 ance, S, are respectively given by 



XA + YB 



XB - YA 



and the impedance, /, is 



/ J 



1 V 



( XA 



. . (9) 



APPENDIX E. 



DISTRIBUTED CAPACITY. 



When electrical energy is transmitted over long distances, the 

 capacity of the cables has to be taken into account. The cables 

 act as an infinite number of condensers in parallel. 

 Let V be the potential at any point P of the cable. 



F" + d V be the potential at a neighbouring point Q. 



i be the current at P. 



i + d i be the current at Q. 



p be the resistance per unit length of the cable. 



C be the capacity per unit length of the cable d x be an 



element of length and d t an element of time. 

 Then 



therefore 



, 



pdx 



dV 



dx 



-pi 



(1) 



