Contributions to the Theory of Alternating Currents. 349 



currents can be represented as simple sine (or cosine) functions of 

 the time. 



The method consists in applying the fact that if a simple harmonic 

 function is differentiated twice in succession the result is propor- 

 tional to the original function. By the application of this principle 

 the determination of the steady values of the currents is reduced to 

 solving a set of simultaneous simple equations. 



After introducing the method by solving some simple problems, it 

 is applied to the following : — 



(a) The determination of the equivalent resistance R, reactance 

 and impedance I, of a parallel circuit of n branches, taking into 

 account mutual induction, when each branch may contain resistance, 

 capacity, and self-induction. 



The result is written 



n A C A B A 



**> ~ C 2 +/B 2 ' u ~ C 2 +p 2 B 2 ' 1 - v /C 2 + i . 2 B 2 ' 



L being the equivalent self-induction, where A , 0, B are certain 

 functions of the resistances, self-inductions, capacities, and mutual 

 inductions of the several circuits. 



(b) The determination of the currents in the n circuits of an air 

 core transformer having one primary coil and n — 1 secondary coils. 



In addition to solving the problem, the conditions for resonance in 

 the primary circuit are obtained and discussed, and special attention 

 is given to the case of a transformer having only one secondary coil. 



(c) The determination of the outputs of n alternators working in 

 parallel on a non-inductive external circuit. 



Paet II. 



This part is devoted to the consideration of the effects of higher 

 harmonics in E.M.F.s and currents on the values of the impedances 

 and reactances of circuits. 



The problems considered in Part I are again discussed on the 

 assumption that the impressed potential difference is of the form 



E = Ei sin (pt— 0i) +E 2 sin (2pt— 6> 2 ) -f . . . +E OT sin (mpt—0 m ). 



It is also shown that periodic E.M.F.s and corresponding currents 

 can in all cases be represented by simple sine curves having the same 

 root mean square values, and suitable phase positions depending on 

 the time constants of the circuits and on the periodicities of the 

 harmonics present. 



2 c 2 



