452 Mr. R. S. Broiigh on the best Resistance 



certain constant, which may be called the " retardation cha- 

 racteristic " of the line, and the expression for which is 



where k is the resistance and c the capacity of the line per 

 mile, and I is the length of the line in miles. 



Now we see that the value of the RC increases as the square 

 of the length of the line ; and since by increasing the resist- 

 ance of the receiving-instrument we virtually increase the 

 length of the line, it is perfectly obvious that, if we make the 

 resistance of the receiving-instrument unduly high, we may 

 increase the value of the RO to such an extent as to impair 

 the signalling-speed of the line. 



It thus becomes clear that in the case of a very long and 

 liiglily insulated line the best resistance for the receiving- 

 instrument, as indicated by the result obtained by examining 

 the problem under the first aspect only, may be so great as to 

 retard the speed of signalling. 



I shall here consider only the case of a perfectly insulated 

 line. 



Let I = the length of the line in miles, 



h = resistance per mile in ohms (supposed uniform), 

 c = capacity per mile in farads (suposed uniform), 

 and r = the resistance in ohms of the receiving-instrument. 



Then the sensibility of the receiving-instrument is 



M= const. 



v/ 



+ kl 



And assuming that the intercalation of the receiving-instru- 

 ment of resistance r in circuit has approximately the same 

 influence on the signalling-speed as increasing the length of 



the line by -7 miles, we have 



^ / rV 



kc{l+-^) 



RC= const. X ^ log^ (f). 



Now, if it be assumed that the efficiency of the receiving- 

 instrument varies directly as its sensibility, but inversely as its 

 retardative influence, then we have the following expression 

 for the efficiency, namely 



RE = const. X 



const. X 



1T^\/ T 





(/•+wy 



3' 



