If k1 and K2 are constant, we have from (1), (5) and (6) d²X² / 1 1 \ ---- = 8[pi]e(q - [alpha]n1n2)( --- + --- ) (7), dx² \ k1 k2 / an equation which is very useful, because it enables us, if we know the distribution of X², to find whether at any point in the gas the ionization is greater or less than the recombination of the ions. We see that q - [alpha]n1n2, which is the excess of ionization over recombination, is proportional to d²X²/dx². Thus when the ionization exceeds the recombination, i.e. when q - [alpha]n1n2 is positive, the curve for X² is convex to the axis of x, while when the recombination exceeds the ionization the curve for X² will be concave to the axis of x. Thus, for example, fig. 11 represents the curve for X² observed by Graham (_Wied. Ann._ 64, p. 49) in a tube through which a steady current is passing. Interpreting it by equation (7), we infer that ionization was much in excess of recombination at A and B, slightly so along C, while along D the recombination exceeded the ionization. Substituting in equation (7) the values of n1, n2 given in (3), (4), we get _ _ d²X² | [alpha] / k² dX²\ / k2 dX²\ | / 1 1 \ ---- = 8[pi]e |q - -------------- (1 + ----- --- ) (1 - ----- --- )| ( --- + --- ) (8). dx² |_ e²X²(k1 + k2)² \ 8[pi] dx / \ 8[pi] dx /_| \k1 k2 / [Illustration: Fig. 11.] This equation can be solved (see Thomson, _Phil. Mag._ xlvii. P. 253), when q is constant and k1 = k2. From the solution it appears that if X1 be the value of x close to one of the plates, and X0 the value midway between them, 1 X1/X0 = ------------------ [beta]² - 2/[beta] where [beta] = 8[pi]ek1/[alpha]. Since e = 4 × 10^-10, [alpha] = 2 × 10^-6, and k1 for air at atmospheric pressure = 450, [beta] is about 2.3 for air at atmospheric pressure and it becomes much greater at lower pressures. Thus X1/X0 is always greater than unity, and the value of the ratio increases from unity to infinity as [beta] increases from zero to infinity. As [beta] does not involve either q or I, the ratio of X1 to X0 is independent of the strength of the current and of the intensity of the ionization. No general solution of equation (8) has been found when k1 is not equal to k2, but we can get an approximation to the solution when q is constant. The equations (1), (2), (3), (4) are satisfied by the values-- n1 = n2 = (q / [alpha])^½ k1 k1n1Xe = ------- I, k1 + k2 k2 k2n2Xe = ------- I, k1 + k2 /[alpha]\^½ I X = ( ------- ) ----------. \ q / e(k1 + k2)