124 Mr. L. Schwendler on the General Theory 



and if P cannot be always kept rigidly equal to Q, we should at 

 least have 



P — Q = S as small as possible, . . . (IV.) 



p, P, and Q being functions of the resistances and electromo- 

 tive forces of the system, which are known so soon as the parti- 

 cular duplex method has been selected. 



The general problem which is to be solved for duplex tele- 

 graphy may now be clearly stated as follows : — 



]) and S are two known functions ivhich 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 ap- 

 proximate rapidly towards zero as the variation in the system 

 becomes smaller and smaller. 



Thus the solution of the problem for any given duplex method 

 will always be a question of the minima and maxima calculus. 



Having then ascertained the best arrangement for each duplex 

 method, the methods can be compared inter se; and that method 

 will be best, and should be selected for use, which for any given 

 variation in the system gives the least absolute magnitude to the 

 functions D and S. 



If we suppose, however, that the particular duplex method is 

 not given, the problem to be solved becomes more general, but 

 would still be entirely within the limits of the variation cal- 

 culus, furnishing, no doubt, a very interesting and important 

 application of that most powerful mathematical instrument. 

 The general solution would at once determine the best method 

 possible, after which special solutions would give the best ar- 

 rangement for that best method. 



It is, however, not my intention to endeavour to solve here 

 the duplex problem in this most general form. To be able to 

 indicate so general and desirable a solution is by no means 

 identical with being able to effect it. The task before me is far 

 more simple, since, as already pointed out, I shall investigate 

 each duplex method separately to determine its best quantitative 

 arrangement, and ultimately compare the different methods to 

 ascertain their relative values. 



To do this, the question may be attacked in two different 

 ways, depending on the purpose for which the solution is 

 required. 



Namely, either the solution is to be made when considering 

 the line as a variable conductor only, but not acting perceptibly 

 as a Leyden jar ; or the line is to be considered as constant in 

 conduction and insulation, but acting as a Leyden jar of large 

 capacity. In the first case the solution would be directly appli- 

 cable to short overland lines (not over 200 miles in length), and 

 in the second case to submarine cables, which, if good, may 



