32 PROCEEDINGS OF THE AMERICAN ACADEMY. 



the receiver. That is, it depends on the " receiving-end impedance " 

 of the circuit, or the ratio of the impressed standard-frequency alter- 

 nating emf. at the sending end, to the current-strength at the receiving 

 end. If this receiving-end impedance of the circuit, including the im- 

 pedance of the receiving apparatus, is not greater than 25,000 ohms 

 (12,500 ohms per wire), at the angular velocity w = 5000 radians per 

 second, commercial telephony will readily be possible with the standard 

 Bell telephone apparatus used in the United States ; unless the dis- 

 tortion of the speech-waves, due to unequal attenuation at differenl 

 frequencies, is unusually great. If the circuit receiving-end impedance 

 exceeds 200,000 ohms (100,000 ohms per wire) at w = 5000 radians per 

 second, even expert telephonists will ordinarily be unable to converse 

 with this apparatus over the line. 



It is easy, with the aid of formulas given in the above-mentioned 

 preceding paper, to find the equivalent n of a simple single telephone 

 line of given length, uniform linear constants, and assigned terminal 

 conditions. But for most practical purposes this is not enough. Most 

 long telephone lines in practical service are not single, but composite. 

 Consider the case of a subscriber A, in Boston, talking to a subscriber 

 B, in New York. First there is the terminal apparatus at A ; then, 

 say, a few kilometers of underground line in Boston. Next comes the 

 long-distance overhead line from Boston to New York, perhaps con- 

 sisting of more than one section and size of wire. Then come one or 

 more sections of underground wire in New York, before we end the 

 circuit in B's apparatus. At two or three intermediate exchanges in 

 this circuit there may also be casual loads, formed by supervisory re- 

 lays, or other instruments. The critical receiving-end impedance must 

 not be exceeded in this composite circuit, if the talking is to be of sat- 

 isfactory quality. Actual trial of the line by conversation will deter- 

 mine, with a fair degree of precision, whether the limiting permissible 

 receiving-end impedance has been exceeded by the line. But the de- 

 signing telephone engineer seeks to know, in advance, whether a certain 

 projected composite line will, when constructed, fall within the per- 

 missible limit of receiving-end impedance. If working formulas can be 

 developed, that are not too lengthy and complicated, for determining 

 the receiving-end impedance of composite lines, they may help the 

 designing engineer to decide questions of line construction. 



In this paper the discussion will be principally confined to direct- 

 current composite lines. The formulas thus derived are all easily 

 presented, grasped, and checked by Ohm's law, since they involve 

 only real numerical quantities. In the direct-current case the hyper- 

 bolic quantities used are all functions of simple real numerics, for which 



