of Air by Ph osph or us . 483 



initial potentials ; but these (about 40 volts) are nearly 

 constant. 



6. Working Hypothesis. — I shall next endeavour to account 

 in some theoretical way for the observations just described, 

 seeing that the data as a whole are very satisfactory when 

 the inherent difficulties (fluctuating ionizer, colour criteria, 

 &c., as repeatedly detailed) are taken into consideration. The 

 method pursued will be alike in principle to that already 

 applied to tubes in the absence of an electric field, in my 

 second paper. The complete equation is to be deduced and 

 the numerical reasonableness is then to be tested when the 

 decay of the ions within the ionized region is ignored. 



Let k be the absorption-velocity, so that ka particles are 

 absorbed per second per square centim. by any barrier, n 

 being the number of particles per cubic centim. As usual, if 

 but J of the particles travel in any given cardinal direction, 

 k is to be replaced by 3k. Let k'n 2 particles decay within a 

 cubic centimetre per second, as the result of mutual destruc- 

 tion* or otherwise. Let v be the longitudinal velocity of the 

 current of air within the tube conveying the emanation through 

 the condenser, eventually to discharge it into the colour-tube. 

 Finally, let r 1 and r 2 be the internal and external radii of the 

 condenser. Consider the element of volume between two 

 right sections of the condenser dl apart. The accumulation 

 within the element is — Tr{r. 2 2 — r^)v(dn[dl)dl\ the loss by 

 decay within the element is per second k / n 2 7r(r 2 2 — r l 2 )dl ; the 

 absorption at the walls (internal and external) of the element 

 is per second k^Trfa + r^dl. Hence in the stationary con- 

 dition 



- v(dn/dl) = k'n 2 + 2nk/(r 2 -r { ). 



This equation is integrable in finite form. To determine the 

 constant of integration, let » be the saturation at the right 

 section l of the condenser. Then 



2kn /n = (2k + k\{r 2 ^r 1 ))e 2k ^^ v ^^' ) -'k f n {r 2 -r 1 ) , 



an equation containing n in terms of the variable I and the 

 parameters k, k', n , v, r 1} r 2 , l . If r^O and r 2 =r, the 

 equation in an earlier paper is again deduced. 



7. Decay ignored. — I will now continue in the manner 

 usual in this investigation, and write ^^O. Then 



n = n /6 2k «- l oy^- r >\ 



* Note that if decay is absent, ions may be supposed to be separated 

 out of the nuclei by the stress of an electric field, and not to occur in the 

 absence of the field. 



