Telephone Circuits. 675 



have only a few entries. Bat calculation is easy enough. 

 Write r{6) in the shape ot + fti. Then 



cosh r(0) = cosh a . cos ft -f i sinh a . sin/9 

 and 



sinh r (6) = sinh a . cos ft + i cosh a . sin /3, 



so that the calculation is easy for any values of a and ft. 



I will first make a few statements, the foundation of my 

 reasoning. 



1. It is well known that any looped line or cable con- 

 taining a receiving instrument at the end of the loop has its 

 equivalent in a cable whose far end goes to earth through 

 fhe receiving instrument, the outer coating of the cable 

 being everywhere to earth or of zero potential. The mathe- 

 matical proof of this is very easy and 1 shall not give it, 

 although I do not know that it has been published. 



2. In all difficult problems cables are so long that for all 

 practical purposes we may suppose them to be infinitely long. 

 Thus it is well known to practical people that when lines of 

 30, 40, &c. miles of the same cable are experimented upon, 

 the rate of diminution of current per mile is the same except 

 for regions near the ends, and these end effects are much the 

 same for all lengths of line except in short lines. 



3. In any long uniform cable at equidistant points A, B, 0, 

 &c, when a simple periodic current is flowing, its value at 

 these points being C A , B . Cc, &c, the ratio of C A to Cb is 

 the same as that of Cb to 0c, &c. This is true except near 

 the ends. 



4. In a long cable with perfectly similar contrivances, 

 AiA 2 , B^o, CxC 2 , &c, at equidistant places A, B, C, &c. ; 

 when a simple periodic current is flowing, its value at corre- 

 sponding points being A u A 2 ; B b B 2 , ; Cj, C 2 , <fcc, the ratio 

 of Aj to B t is the same as that of B, to Cj, &e., and these 

 are also the same as the ratio of A 2 to B 2 , B 2 to C 2 , <fcc. 

 We may say the same about the voltages at corresponding 

 points. It is seen that I assume an infinite line, but even in 

 the case of an infinite line my assumption is that there is 

 some diminution in the amplitude of the current from A t 

 to B t . If the circuit behaved like a mere Ohmic resistance 

 the current would be the same everywhere but the voltage 

 would not. 



5. Neither telephone nor telegraph signalling can be 

 studied correctly through the behaviour of periodic currents, 

 but it has been found in practice that in telephony, if we 

 assume that all currents are periodic functions of the time, 

 the frequencies being not much less than 600 nor much 



2X2 



