1877.] Telegraf% Beeeiving Instrmnent. 187 



Then the sensibility of the receiving instrument is : 



M = Const. X , , 



r + k I 



And assuming that the intercalation of the receiving instrument of 



resistance r in circuit has approximately the same influence on the signall- 



ing speed as increasing the length of the line by - miles, we have 



Jcc(l-\- y) 

 RC = Const. X —^ 



■"«.©■■ 



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

 ment varies directly as its sensibility, but inversely as its retardative influ- 

 ence, then we have the following expression for the efficiency, namely : 



EE = Const. X ^ Vr 



= Const, 

 which is a maximum for 



hc(l + ^\ (r + lcl) log^ (f ) 



h I 



that is, the resistance of the receiving instrument in the case of a perfectly 

 insulated and uniform line should be one-fifth of the resistance of the hne.* 



Taking into consideration the resistance of the signalling battery, 

 which has hitherto been neglected, the result is modified as follows. 



Suppose we are given a certain number of cells (all of equal electro- 

 motive force and resistance) and arrange them so that the total resistance 

 of the battery = /, then it may easily be shewn that the total electromo- 

 tive force of the battery will be proportional to \/ f 



Thus the expression for the sensibility of the receiving instrument be- 

 comes (employing the same notation as before) 



M = Const. X . y^^. , 

 /+ r + hi 



and the expression for the retardation characteristic becomes 



/+rN 



Ic c (I -{■''—^\ 



EC = Const. X 2 — "^ log (o) 



and finally, the expression for the receiving efficiency of the instrument 

 becomes, 



* Singularly enough, this is the precise value selected, on experimental grounds, 

 by Prof. Hughes. 



