Oscillation- Hysteresis in Simple Triode Generators. Ill 



Setting dx fl =(p l , 0, 0, 0) we see by (5) that the equation 

 is satisfied; that is to say, p x is the radius of the quadric in 

 the x direction. The quadric is seen by inspection of its 

 equation to be invariant: consequently we may take x x in 

 anv direction we please. Hence the radius of the quadric 

 (6) in any direction is equal to the radius of (spherical) 

 curvature of the corresponding section of the world. 



If G fJiV = \g fXV , then G = 4\, and the quadric reduces to 



— Xg^dx^dx,, = 3 



or —ds 2 = 3/\, 



showing that the quadric is a sphere of radius ^/(3/X,) . 

 Conversely, if the radii of spherical curvature of sections of 

 the world at all points and in all directions are equal to 

 \/(3/A,), Einstein's equations G^ = Xg^ will be satisfied. 

 This demonstrates the theorem. 



It may be noticed that in this general prooE we have 

 substituted spherical curvature for the normal curvature 

 considered in the Appendix to the previous paper. This is 

 necessary because in the general case normal curvature 

 becomes meaningless. 



XVI. On a Type of Oscillation- Hysteresis in a Simple Triode 

 Generator. By E. V. Appleton, M.A., M.Sc, Fellow of 

 St. Johns College, Cambridge, and Balth. van der Pol 

 junr., JD.Sc, Conservator Physical Laboratory of Teylers 

 Institute, Haarlem (Holland^) *. 



THE conditions for the production of free infinitesimal 

 oscillations in various triode circuits have been worked 

 out in great detail during the last few years, but the question 

 of the stability and maintenance of oscillations of finite 

 amplitude does not appear to have received equal attention. 

 In a recent paper f we have dealt with the calculation of the 

 amplitude finally attained in a simple case of free triode 

 vibrations in which use was made of a non-linear " oscillation 

 characteristic" easily determined by experiment for any 

 particular tube and circuit. This oscillation characteristic, 

 which represents the relation between the variations of anode 

 potential and of anode current, may be regarded as expressing 

 the electrical properties of an imaginary non-reactive re- 

 sistance connected in parallel with the inductance of the 



* Communicated by Professor Dr. H. A. Lorentz, For.Mem.RS. 

 t Phil. Mag. vol. xlii. p. 201, August 1921. 



Phil. Mag. S. 6. Vol. 43. No. 253. Jan. 1922. N 



