470 Mr. L. Schwendler on the General Theory 



.p p v a 



q '4a~+ZL + b' 



which has an absolute maximum with respect to a only ; namely 

 <L b 



Substituting this value of a in the last expression for P, we get 

 p= Eg 1 



Whence it follows that P becomes largest for b = ; otherwise 

 b remains indeterminate, q, on the other hand, should be made 

 as large as possible. 



r 

 If we now put v— — n and develop its value from the balance- 

 equation, we get 



q 2 V 2L-H 



The solution of the first problem of the differential method, 

 when the line is perfect in insulation, is therefore 

 h=d=0 3 

 f=b = w + {3, 

 L b 



2L + 6* 



The absolute value of b is left indeterminate*, and we only 

 know that the smaller it can be made the better. 



But to fulfil this best condition, f^b = w + /3 = Q represents a 

 physical impossibility, since neither /3, the internal resistance 

 of constant galvanic cells, can be made zero, not even approxi- 

 mately, nor b } which must have convolutions in order to act mag- 

 netically. 



The larger f=b = w-\-/3 becomes, for practical reasons, the 

 more the differential method, even under the best quantitative 

 arrangements as given above, will become inefficient as compared 

 with the double balance. 



Now by inference we get for a line with leakage, i. e. i <oo , 



* Practically, however, it may be said that b is given ; for generally /3, 

 the internal resistance of the signalling-battery, is determined by the 

 nature and number of galvanic cells required for duplex working. We 

 must only remember that b should be made somewhat larger than /3, in 

 order to have an adjustable resistance w in the battery branch, which may 

 be used for compensating any variation of the battery-resistance, that the 

 equation f—b—w-\-/3 may be permanently fulfilled. 



