Quantum Theory of Electric Discharr/e. 235 



or of the form a + 6E . The minimum spark potential must 

 be then independent of the pressure. This has been found 

 to be the case. 



4. Now, it is well known that as the pressure of the gas 

 decreases, a dark space is formed next to the cathode called 

 the Crookes dark space. And it is obvious that until the 

 energy of the ionizing corpuscle attains to the minimum 

 value required, it will not be in a condition to ionize. This 

 explains the formation of the Crookes dark space, and 

 further suggests that the difference of potential between 

 the cathode and the negative glow, called the cathode-fall 

 of potential, must be approximately equal to the minimum 

 spark potential and independent of pressure. This is also 

 in accordance with known experimental results. 



As the electric intensity in the Crookes dark space is 

 very high, it will follow that the breadth of this space 

 will be very small, increasing as the electric intensity 

 decreases with pressure. It is seen further that the cathode- 

 fall of potential must be proportional to E , or at any 

 rate be of the form a^ &iE . 



5. Again, the energy of a corpuscle when it collides with 

 a molecule of the gas is evidently proportional to JLel, 

 where X is the electric intensity and I the mean free path. 

 Hence Xe/ = E. Moreover, it can be shown that if a. is 

 the coefficient which determines the rate of increase in 

 the number of corpuscles, then, when a is equal to zero, 

 ~Kel is constant. We find now that this constant value 



must be =E . Since, further, Zac — where p is the pressure, 



X 



we find that when a is zero, — = const., a result which was 



P 



obtained otherwise in a previous paper (Phil. Mag. Oct. 

 1912). 



6. Returning now to the Crookes dark space, we observe 

 that it is limited by the point at which, through the action 

 of the electric force, the energy = E , and, accordingly, 

 this also marks the beginning of the negative glow. As 

 the corpuscle moves through the negative glow, its energy 

 decreases on account of collision, according to the law 

 given (as J. J. Thomson has shown) by E 2 =E 2 — 2fix 



or E = E (l—~^x) nearly, where x is the (small) distance 



