72 Mr. O. Heaviside on the 



Suppose branches m and n are to be conjugate, so that an 

 E.M.F. in m can cause no current in n. First exclude m's 

 equation from (24c/) altogether, and, with it, Z mm . Then 

 write down the equations of E.M.F. in all the independent 

 circuits of the remaining branches, by adding together equa- 

 tions (24cZ) in the proper order ; this excludes the V's, and 

 leaves us equations between the «'s and all the independent 

 C's, but one fewer in number than them. Put the G m terms 

 on the left side, then we can solve for all the currents (except 

 C OT ) in terms of C m and the e's. That the coefficient of C m in 

 the C n solution shall vanish is the condition of conjugacy, and 

 when this happens, C n is not merely independent of e m 

 but also of Z mm , though not of Z ral , Z m2 , &c. 



I have dwelt somewhat upon this property, and how to 

 prove it for transient states, because, although it is easy 

 enough to understand how the current in one of the conjugate 

 branches, say n, is independent of current arising from causes 

 in the other conjugate branch, m,, yet it is far less easy to 

 understand how, when m is varied in its nature, and therefore 

 wholly changes the distribution of current in all the branches 

 (except one of the conjugate ones) due to impressed forces in 

 them, it does not also change the current in the excepted 

 branch n. Conscientious learners always need to work out 

 the full results in a problem relating to the steady-flow of 

 current before they can completely satisfy themselves that the 

 property is true. 



Note on Part III. Example of treatment of terminal con- 

 ditions. Induction- Coil and, Condenser — One of the side- 

 matters left over for separate examination when giving the 

 main investigation of Parts 1. to IV . was the manner of 

 treatment of terminal conditions when normal solutions are 

 in question, especially with reference to the finding of the 

 terms in the complete solution arising from an arbitrary 

 initial state which are due to the terminal apparatus, concern- 

 ing which I remarked in Part III. that the matter was best 

 studied in the concrete application. There is also the ques- 

 tion of finding the nature of the terminal arbitraries from the 

 mere form of the terminal equation, without knowledge of 

 the nature of the arrangement in detail, except what can be 

 derived from the terminal equation. 



Let, for example, in the figure, the thick line to the right 

 be the beginning of the telegraph-line, and what is to the 

 left of it the terminal apparatus, consisting of an induction- 

 coil and a shunted condenser. The line is joined through the 

 primary of the induction-coil, of resistance P T , to the con- 

 denser of capacity S , whose shunt has the conductance K , 



