Triode Generator with Two Degrees of Freedom. 703 



■contributions to the problem have already appeared *, the 

 phenomenon has, as far as we are aware, only been dealt 

 w T ith in a linear theory. The solutions of the differential 

 equations in this case are of the form tf ±a *sino)£, and it 

 depends on the sign of a. whether an oscillation will build 

 up or decay. But whether both oscillations will be present 

 simultaneously or whether the one mode of vibration will 

 suddenly be replaced by the other when a pans meter of the 

 circuits is gradually varied, and whether a hysteresis loop 

 will be obtained, these questions can not be answered by a 

 linear theory. In order therefore to retain in the analysis 

 the necessary interaction of simultaneous vibrations which 

 determines the stability of the oscillat ons, non-linear terms 

 which occur through the curvature of the triode charac- 

 teristics may not be ignored. 



Before attempting, however, to set out a non-linear theory 

 -of the phenomena under consideration, a few remarks may 

 first be made concerning the terminology. 



The notion of one or two degrees of freedom is used here 

 as an extension of the usual meaning attached to these terms 

 in the ordinary linear treatment of oscillation problems. 

 We are wed aware that, e.g. to speak of a system as havino- 

 one degree of freedom when more than one stable oscillation 

 is possible for a given set of parameters f is not altogether 

 satisfactory, but it is hoped that from the description of the 

 phenomena the meaning will bo sufficiently clear. 



Further, we shall discriminate between a " possible" vibra- 

 tion and a vibration that can actually be realized. With 

 " possible " is here meant a solution representing a stationary 

 oscillation with a constant amplitude. It may, however, be 

 that this oscillation cannot be realized, it being unstable. 



Finally, it is obvious that for a system such as shown in 

 tig. 1, when the secondary circuit is very loosely coupled to 

 the primary, the reaction of the secondary on the primary 

 will be small. It is found experimentally that under these 

 circtimstances an ordinary resonance curve can be obtained 

 as th»* secondary circuit. This case will, however, not be 

 considered here and e shall confine our considerations to 

 cases where the coupling is strong. 



* J. S. Townsend, Radio Review, i. p. 369 (May 1920). K. Heegner, 

 Archix, fur Elektrotechnik. ix. p. 127 i!920). F. H-<rn*s, Jahrbuch fur 

 drnhtl. Telegruphie, xv. p. 442 (1920). fl. Vogel und M. Wien, Ann. d. 

 Phys. lxii. 'p. 649 (1920). H. G Moller, Jahrbuch fur drahtl. Tele- 

 graphic, xvi. p. 402 (1920). H. Pauli, ibid. xvii. p. '322 (1921). W. 

 R<»D-owski, Archie fur Elektrotechnik, x. pp. 1, 15 (1^21) See also 

 Moller, Die Elektronenrohre (Vieweg, 1920). 



t See, e. g. Appleton and Van der Pol, Phil. Mag. Jan. 1922. 



