KENNELLY. — ARTIFICIAL LINES FOR CONTINUOUS CURRENTS. 115 



thereby be reduced from 100 to 64.343 volts. The effect of this will 

 be to reduce the received current at B to 0.01591 ampere, the same 

 as in Figure 7. 



Best Resistance of Receiving Instrument. 



Electromagnetic receiving instruments may be divided into two 

 classes; viz. (1) those, as of the movable-coil type, in which the mag- 

 neto-mechanical force, or torque, is directly proportional to the ampere- 

 turns, and (2) those, like simple non-polarized relays, in which the 

 magneto-mechanical force, or torque, may be nearly proportional to 

 the square of the ampere-turns at low magnetic saturation, but, as 

 saturation increases, to perhaps a lower power of ampere turns than 

 the first. In either case, the magneto-mechanical force may be ex- 

 pressed by: 



F = a (I ni) p dynes or dyne-perp. cms., (76) 



where F is the force or torque, a is a constant of the instrument, I is 

 the received current in amperes, n 1 the number of turns in the winding, 

 and p some exponent not greater than 2. The received current I is 

 expressed by (42) or (56). The number of turns n x in a given winding 

 space is well known to be sensibly proportional to vV, where a is the 

 resistance of the winding in ohms, provided that the size of copper 

 wire selected is within the fairly wide range that keeps the ratio of 

 covered diameter to bare diameter sensibly constant. Consequently, 

 we have approximately: 



F = a' ( r-r— - — ~ 7-t — 2 J dynes, or dyne-perp. cms. 



\r sinh 2 md + <r cosh 2 m6 ) J J r tr 



In order to make this force a maximum by varying <r, we differentiate 

 F with respect to <x in the usual way, and equate to zero. We then 

 obtain 



a- — r tanh 2 mO — r tanh La ohms. (78) 



That is, the best resistance for the electromagnetic winding of the re- 

 ceiver is equal to the sending-end resistance R g of the line, no matter 

 what the exponent p which expresses the relation between torque and 

 ampere-turns. 3 



3 See Ayrton and Whitehead paper in Bibliography. 



