enu = Cε(α−γ)x − γi / (α − γ), emw = -Cε(α−γ)x + αi / (α − γ),
where C is a constant of integration. If there is no emission of positive ions from the anode enu = i, when x = d. Determining C from this condition we find
| enu = | i | {αε (α−γ) (x−d) − γ }, emw = | αi | {1 − ε (α−γ) (x−d)}. |
| α − γ | α − γ |
If the cathode did not emit any corpuscles owing to the bombardment by positive ions, the condition that the charge should be maintained is that there should be enough positive ions at the cathode to carry the current i.e. that emw = i; when x = 0, the condition gives
| i | { αε−(α−γ)d − γ} = 0 |
| α − γ |
or
ε αd/α = ε γd/γ.
Since α and γ are both of the form p∫ (X/p) and X = V/d, we see that V will be a function of pd, in agreement with Paschen’s law. If we take into account both the ionization of the gas and the emission of corpuscles by the metal we can easily show that
| α − γε(α−γ)d | = | kαVe | [ | 1 | − ε−(β+γ−α)d{ | 1 | + | d | }], |
| α − γ | d | (β + γ − α)² | (β + γ − α)² | β + γ − α |
where k and β have the same meaning as in the previous investigation. When d is large, ε(α−γ)d is also large; hence in order that the left-hand side of this equation should not be negative γ must be less than α/ε(α−γ)d; as this diminishes as d increases we see that when the sparks are very long discharge will take place, practically as soon as γ has a finite value, i.e. as soon as the positive ions begin to produce fresh ions by their collisions.