soMr. co.v/r.uroK.iA'K .tnf.iKcrf /.v riivsics- n i.ii 



an iHiuation expressing the dal.i e(niall\' well. If we adopt the former, 

 and desiitnale by (i the niiinher of moleniles ionizetl i)y a positive 

 ion in a centimetre of its path, and by X,> llie iininhcr of electrons 

 supplied [wr second at the cathode, we ^;et 



Of course, ii must he mnili smaller than <*, or tiic posilixe ions would 

 have made themseUes felt earlier. Or if we adopt the latter idea, 

 and tiesiRnate by k the number of electrons expelled from the cathode 

 (on the average) by each positive ion striking it, we arri\e at the 

 formula 



^^ 1 -*(«•«■'- !)• ^'' 



N'atur.dl>- k must be imich smaller than unity for the s.iiiie reason. 

 In Fig. r> the broken curve represents (ti), with the values <S. 1(5 and 

 .0067 assigned to a and ji; it also represents (7), with the values 

 S. 10 and .00082 assigned to a and k." (It was expected that the 

 cur\es representing the two equations would be perceptibly apart 

 on the scale of Fig. ."); but they were found to fall iiidistiiiguishably 

 together.) 



Fvidently, therefore, the positi\e ions, weak and lethargic as they 

 are in liberating electrons (one has only to compare with a, or look 

 at the \alue assigned to k in the last sentence!), can produce a notable 

 addition to the current when the electrodes are far enough apart; and 

 more than a notable addition, for when the dis'ance d is raised to the 

 value which makes the denominator of (6) — or of (7), whichever 

 equation we are using — equal to zero, the value of .V is infinite! Per- 



" The derivations of (6) and (7) are as follows. Represent l)y M (x) the nunil)er of 

 electrons crossing the plane at .v in unit time (the cathode being at .v = o and the anode 

 at .v=</); by P (x) the numlwr of positive ions crossing the plane at x in unit time; 

 by .V,, the numtier of electrons independently supplied at the cathode per unit time, 

 which is not necessarily equal to the value of M at .x=o (hence the notation); by i 

 the current, or rather the current-density, as all these reasonings refer to a current- 

 flow across unit area. We have 



Me+Pe = i, hence 

 making the assumption which leads to (6) we have 



dM/dx = aM+0P = (a-/3) M+pi/e 



The boundary conditions are: ..V = A'oat .v=oand M = i''eat x=d. Integrating the 

 equation and inserting these we get (6). Making the assumption which leads to 

 (7) we have 



dM,dx = aM 



The boundary conditions are: M = NQ+k(i/e — Af) or (l-|-it;.l/ = iVo-|-At/< at x = o, 

 and M = i/e at x = d. Integrating the equation and inserting these we arrive at (7). 



