J 2911 



-e,-).v,-ri,. ....... (10) 



Krom (10) we see in coiinectioii with the above discussion tiiat, 

 if we are at tlie proper point of the characteristic (e.g. B iji fig. 3) 

 and the grid is subjected to potential-fhictuations r,, we may consider 

 the audion as an alternating cnrrent-generator with an electromotive 

 force Az^i and an internal resistance r. Tlie potential-liuctnations of 

 the grid may l»e caused by an external electromotive force. But the}- 

 may also be produced by coupling the grid in a proper numner to 

 the anode-circuit, by which an oscillation when once arising, main- 

 tains i<self. Both methods find manifold application in wireless 

 telegraphy. The second method will be especially discussed here, in 

 doing so we shall make use of the method of '"complex resistances", 

 wliich is customary in allernating-current-theory ; the following- 

 remarks on this method may be useful. 



We suppose an arbitrary electric system, consisting of self-induc- 

 tions, capacities and resistances, in which somewhere an electromo- 

 tive force E cos pt is applied. The currents which arise in the 

 system, satisfy a set of linear differential equations') of the form: 



di'ii ri/i dt y 



(It J Ch \ E COS pt 

 where the summation is to be extended over all the currents 

 occurring in any closed circuit which can be described in the 

 system. The solution of (11) consists of the general solution of a 

 set of homogeneous linear ecpiations, which are obtained from (11) 

 by putting li co.'i /d= 0, and a particular solution. The first part of 

 the solution gives the (damped) free vibrations of the system ; the 

 second part the foired vibration. To discover the particular solution 

 use can be made of the complex notation by ()utting E eJi'^ for 

 Ecospt, where j=l —1; and by trying for //, a. solution of the 

 form Ahe.'h'^; Ai, is complex and gives not only the amplitude but 

 also the phase-shift of //,. 



Instead of (11) we thus obtain a set of linear algebraic equations 

 of the form : 



:s|ft.-f,/(pL.-l)j^„ = <(^. . . . (12) 



Equations (12) are analogous to Kirchhoff's equations for a direct- 

 current-system, oidy in the present case complex resistances occur 



of the form Zh = Hi, -\- j \ nLj, — — ). Therefore the same rules may 



■ V pCiJ 



1) At the same time they satisfy algebraic equatioui of the form ^ij. = 0: 



^Rr^ikVU ", ^ y~ =^<^ ,, _. . . . (11) 



