On Tliermionics. 815 



ions received by any of the surfaces in a given time will be 

 invariable. The problem is to find the number of ions which 

 reach any of the surfaces B in a given time, together with 

 their velocity components, when the steady state has been 

 established. In the discussion it will be assumed that the 

 motion of the ions is determined solely by their positional 

 and velocity coordinates when they are emitted and by the 

 electric field. The forces exerted on the ions by each other 

 and by molecules of gas into whose sphere of action they 

 may chance to penetrate are left out of account. These 

 conditions are capable of being realized in practice with 

 close approximation if thermionic currents of moderate size 

 are experimented with in high vacua. In order to avoid 

 complications arising out of recombination we shall also 

 suppose the temperature conditions are such that ions of only 

 one sign occur. 



§ 3. General Discussion in Rectangular Coordinates. 



Let the coordinates of any point on one of the surfaces A 

 be x y z 0) and let an ion be projected with velocity com- 

 ponents ?^ v ic from .i' y z Q . Let us seek the condition that 

 this shall strike one of the surfaces B whose equation is 



+(xyz)=0, (l) 



within an infinitesimal distance of the point ,i\ y 1 z Y . 



If V is the potential at any point of the field the equations 

 of motion will be 



av bv . bv ... 



0^ J dy d~ 



On integration these equations give three equations between 

 .r y z and t involving six arbitrary constants which are de- 

 termined by the values of x y z it v Q w . After elimina- 

 tion of the time there result two equations which may be 

 written 



0i (•*' V ~ #o ,'/o -o "o <\> ">o) = 0, ... (3) 

 02 y 3 .'o 2/o Zq "o i' Wq) = 0. ... (4) 

 The curve in which the surfaces <p x and <f> 2 intersect is the 

 trajectory of the particle projected under the given initial 

 conditions. The intersection of this curve with the surface 

 ^(xy z)=*0 will give the point where the particle strikes 

 the surface. The coordinates Xi y x z 1 of such points will 

 therefore be given by solving (1) (3) and (4) for ,v y and z, 

 and the density of these points on the surface yjr will deter- 

 mine the thermionic current density into this surface in the 

 steady state. 



