Equations occurring in Wireless Telegraphy. 207 



Such equations occur in the theory of the production of 

 sustained electrical oscillations by various types of generators, 

 such as the Poulsen and Duddell arcs, and the Dynatron, or 

 the ordinary type of three-electrode thermionic tube. 



If we put t=— and write p instead of V l5 equation (1) 



takes the form 



m 



d 2 p . f(p)d) 



da>" 



'-^+P=0, ...... (2) 



which w T e shall take as the standard. 



In order to obtain a solution of equation (2) we shall 

 regard p and co as the usual p and co coordinates of a point 

 on a curve ; that is to say, p is to be taken as the length 

 of the perpendicular from a fixed point on a tangent to the 

 curve, and co the angle which the perpendicular makes with 

 a fixed line in the plane. 



Now it is a well-known theorem in the differential calculus 

 that 



ds d 2 p /rtN 



-r =P+T^' ( 3 ) 



dec M doD z y 



where s is the length of the arc of the curve measured from 

 a fixed point in it. 



Substituting in equation (2), we get 



ds fip) dp = q 

 da) m da> ' 



which on integration gives 



*+ ^§ f(p)tp=C, (4) 



where C is some constant. 



This is the p and s equation of any curve whose p and 

 co equation is a solution of the differential equation (2). 



We have now to investigate a method of plotting such a 

 curve. 



Take a piece of transparent squared paper, (squared 

 tracing paper), and taking rectangular axes OX and OY, plot 

 on it in Cartesian coordinates the curve 



f+iJ/W^O (5) 



