EXPERIMENTS WITH IONIZED AIR. 39 



ti. Oompntatinth for saturated emanMlon. — Let -■Zc, figure 2, be tbe thickness 

 ui' ;iii iiiiinitesiraal right section of the absoi])tioii tube of radius r, traversed at 

 velocity v by an air current charged with nuclei. Let n be the 



uunil)er jier cubic centim., and h the number absorbed per square ^ 



centim. per second, if n = 1. Hence h is the absoiption velocity S - ^^ 

 discussed. If, as in cases considered elsewhere, the absorption tube 



— T 

 2r; 



were a condenser with the field acting radially outward (Chapter V.), §' 



I- would be replaced by UeE/{R^-E^), where e is the wpiivalent ''"- = -r'>A™AM. 

 charge of an electron, U the velocity of the ions relatively to each other, 

 l-J/iB.T—R^) the potential gradient, R-^ being the a.vial and Jl.y the circum- 

 ferential radius of the conden.ser. I mention this here for future let'erence. 



Let h' be the numbei' of particles decaying by mutual destruction, etc., per 

 cubic centim. per second, if )h = 1, so that h'?i- is the number vanishing for the den- 

 sity of distribution ii. 



Hence the number of particles accumulating [ter second in the element is 



— ■7rr'^v(dn/dx)<l(V^' the number absorbed per second by the walls of the tube, hi2 

 TTi'iIx; the number decaying [)er second within the e\en\%x\i, h'ti-T-r'^dx. Thus 



— (y/^') (dn/dx) =■ 'ikn/h'r + //''. This equation is iutegrable in finite form, and 

 putting OTo as the concentration at a?o, the equation becomes w = 2/?"rto/(£^''''"-^°''''''" 

 {2h -\- h'rn^^ — h'nif). The direct discussion of this equation is cumbersome. Its 

 bearing on the present results is best shown by evaluating the two s[)ecial cases in 

 which h' = 0, and h = 0, I'espectively. The former case is incompatible with the 

 observations and may be dismissed. See § 9. 



7. Case of ahsorption without decay. — Let, then, ^''^ 0, so that decay within 

 the element from any causes whatever is absent.' The only loss of nuclei is at the 

 surface of the absorption tube.- Hence, n = Wot '"'■-'"", if v^o is tbe concentration at 

 a-Q = 0, i. e., in the absence of the absorption tube. But v = 1000 V/60-n-r^, if V 

 liters per minute produce the velocity v centims./sec. Hence, » = ^i, e "'^''•^'''•^^^ 

 The total number of nuclei injected into the color tube is thus n V. Let these pro- 

 duce the fiducial clear blue field. In the same manner let n' V' nuclei produce the 

 same field when the dimensions of the absorption tube are r' and x\ and the air 

 passing V liters per minute. Then, since n V— n' V, Ve -*-/-«5'' _ y ^-ir-.'/..6sy'^ 

 If Fo be the volume pei' minute when the tube length is «' = and the field the 



identical blue, 



k=2.65{J7rx)ln{V/F,), 



an equation from which the value of k, the velocity of the nucleus, is computed at 

 once in cm./sec, supposing decay (k') to be a vanishing quantity. 



Waiving the more refined methods of the kinetic theory of gases, if but ^ of 

 all the nuclei wander in a given direction, the term expressing absorption of the 

 wall of the tube in the differential equation would be h(n/S)2Trrdx, or 7i/S replaces 

 k. Hence the data in the above tables should be inci'eased threefold to meet this 

 point of view, as stated in the first paragraph of this paper. 



' More generally the decay is compensated by the production of new ions within the region by 

 collision.s, etc. 



