of Duplex Telegraphy. 109 



the resistances of the different branches of the bridge arrange- 

 ment — under the limiting supposition, however, that the line 

 used for duplex working was perfect in insulation, or, more gene- 

 rally, that the real conduction-resistance of the line could be neg- 

 lected against the resistance of the resultant fault*. 



It now remains, therefore, to investigate if the simple rela- 

 tions given are generally true ; or if not, what they become in 

 case the line has an appreciable leakage. In fact this is clearly 

 the case of practical importance; since all overland lines, especi- 

 ally long ones, even if constructed on the best known principles, 

 will always have a very considerable leakage; i. e. the resistance 

 of the resultant fault (i) will generally be by no means very large 

 in proportion to the real conduction-resistance (L) of the line. 



In order to obtain the best general solution of the problem, 

 we must conduct the investigation with great caution ; that is, 

 we must be careful not to introduce beforehand any relation be- 

 tween the different variables, however convenient, that is not 

 necessarily a consequence of the paramount condition to be ful- 

 filled for duplex telegraphy, i. e. regularity of signals. 



Thus it will be seen that the present general investigation 

 must be conducted somewhat differently from the special one 

 given in the First Part. 



It must, however, be understood from the beginning that, 

 whatever the best relations may be which should exist between 

 the* different resistances of the bridge method when used on an 

 imperfect line, these relations must revert to the special one 

 given before if we put i = oo ; and this fact affords a certain check 

 upon the correctness of the new relations to be found. 



General solution of the first problem for the Bridge Method. 



The annexed diagram (p. 110) represents the general case; 

 and to it therefore I shall refer in the present paper. 



The general mathematical question which is to be solved for 

 duplex telegraphy has been stated as follows : — 



Regularity of Signals. — D and $ are two functions which 

 must be rigidly equal to zero when no variation in the system occurs 

 — and which for any given variation in the system must be as small 

 as possible, and approximate rapidly towards zero as the variation 

 in the system becomes smaller and smaller. 



Further, these two functions D and S were expressed, say for 

 station I., as follows : — 



E"iNV my K > 



* For a definition of the terms " resultant fault," " real conduction," 

 " measured conduction," " real insulation," " measured insulation," &c, 

 which will be of frequent occurrence in this paper, see my ' Testing In- 

 structions,' Part II. Section I. 



