EXPERIMENTS WITH IONIZED AIR. 67 



second, as the result of mutual destruction * or otherwise. Let v be the longitudinal 

 velocity of the current of air within the tube conve^-ing the emanation through the 

 condenser, eventually to discharge it into the color tube. Finally let Vj and y\, l)e 

 the internal and external radii of the condenser. Consider the element of volume 

 between the two right sections of the condenser, c//, apart. The accumulation 

 within the element is — 7r(/'| — r^) v (dn/dl) dl ; the loss by decay within the ele- 

 ment is per second h'/i^n (^r'i — r^^) dd ; the absorption at the walls (internal and 

 external) of the element is per second kn.27r (/'^ + '"i ) dd- Hence in the stationary 

 condition, —v{dn/dl) = l'')i'^ -\-2nh/{r^—r^). This e(piation is integrable in finite 

 form. To determine the constant of integration let n^^ be the saturation at the 

 right section l^^ of the condenser. Then 



2hnjn = (21 + /•'/., (^'^-'-i) ).«*<'-'.■''"■<'•-'•'' -k'n,(r,-r^), 

 an equation containing n in terms of the variable I and the pai'ameters, k, k', n^, 

 V, ?',, y,, /(,. If /', = and /'g = /•, the equation in Chapter III. is again deduced. 



7. Decay u/nored. — I will now continue in the manner usual in this memoir 

 and write h' = 0. Then n = H.^yg^wf-zj/ccr,-*-,)^ where /^„ is the mimber of particles 

 at the right section /^ of the condenser, and ii. the number at /. Thus /i depends 

 essentially on h and is not, as in the preceding and following chaptei's, independent 

 of it. Moreover h is to be replaced by .'5^ as ali'eady intimated, the latter being 

 nearly identical in numerical value with ?/of the next paragraph. 



8. Infect of an electric field. — Supposing this number of particles to be actually 

 conveying electric current, oi', from a diffei-ent point of view, supposing that one ion 

 is separated in case of the phosphoi'us emanation from each of these nuclei by the 

 stress of the electric field, the current may be found as usual. Oi- the other theory 

 developed in my last pai)er may l)e applied, remembering that the effect of the 

 presence or absence of the electric field on the number of nuclei is inappreciable as 

 demonstrated by the color tube. 



Let E be the difference of potential of the faces of the condenser (potential 

 of the inner rod since the metallic envelope is continually earthed), JJ the velocity 

 of the positive ions relative to the negative ions in the unit field, Q the charge of 

 the condenser, its capacity. Then the chai'ge per second at the element dl is 

 2tt EUne (r, +?',) dl/ii'^ — r^) in consequence of radial conduction across the 

 ionized air space. The total loss of charge per second is dQ/dt = CdE/dt, whence 



«'■=-'•') J lo ^ ■ 



If now the value of n = n^ ^-^((-^oV'Xr.-r.) j^g^ f^uud for k' = is inserted and the 

 integration completed, the result will be 



- (dE/dt)/E^ '''''+l''a''"'' (1 - f -2*=<'-'«'/^«-=-'-''). 

 It is next necessary to evaluate the velocity v (cm/sec) in terms of the volume V 

 (litres/minute), taken out of the gasometer and passed through the condenser into the 

 color tube. Since V 1000/60 = n: ('r| -rf ) v, the datum V if inserted into the last 

 equation reduces it eventually to, 



' 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. 



