of Duplex Telegraphy, 469 



These two expressions do not as yet contain the balance-con- 

 ditions. 



The factors u! _ u," 



and 



2{a" + c") + b" 2{a' + d) + V 



are identical, namely 





2(a" 4- c") + b" " 2(a' + c') + b' " Q 

 where 



Q = e{2(a' + a" + /' + Z'0 +# + #'} +^ 



+ («" + Z") (a' + ? + 6') + (a' + /')(«" + /" + 6"), 



as can be easily calculated by substituting for X, a, and c their 

 known values. 



In the second investigation it has been stated why P' and P" 

 cannot be made maxima separately, and that we could do nothing 

 else but make their sum a maximum. In this case we have 

 to do the same. Hence the question to be solved is reduced to 

 the following : 



p-F+y-,-.™; 18 *' 



is to be made a maximum with respect to the variables a, b } q, 

 and r, while they are linked together by two condition-equations, 

 namely 



r' (a' +c') — q 1 \/ a! b' = balance in station I., 

 and 



r"(fl" + c") -fx/aW f =0 „ „ II. 



This general problem can be solved in exactly the same way 

 as it was in the second investigation. However, it is not 

 needed to do this again, since the general solution can be written 

 down from inference, after having solved the special problem for 

 a line which is perfect in insulation. 



Suppose that i — co , or at least very large as compared with 

 Z' + ^" = L, then obviously P' and P" become identical without 

 condition; namely, 



p — p/z — p— jjL 2q\/a + r\/b . 

 while the two balance-equations become also identical, namely 



2qV ab-r^a + b + 2L) = 0. 



If we substitute the value of r from the balance-equation in 

 the expression for P, we get 



