EXl'EKJLMENTS WITH IONIZED AIR. 83 



The values so found nve then constructed in the chart, figui'e 11. It ai>[)ears 

 at once tliat the data as a whole, though investigated with care, still fail to lend 

 themselves for the nice discernment of the nature of the locus in a relation of cur- 

 I'eiit, 6!</(U, to the diameter 27? of the condenser. The results even of a single 

 series are not smooth. Indeed the exceptional positions of the I'esults for the 

 standard condensers, those of KG being abnormally high while those of K4 are low, 

 is perple.ving and has led me to suppose that some occult cause of variation has been 

 left undiscovered. The relation suggested is really sinuous. One is almost tempted 

 to infer that each condenser behaves as an individual, a conclusion for which I am 

 unable to assign ade(pKite rea.sons. The curve which has been put through the 

 observations was computed from (d V/dt)^^(E -\- a) — A, \n the way presently to 

 be explained.' The observations are in accord with it, in so far as they show an 

 increase of current at an acclerated rate as diameter decreases. 



8. Worhivg Injpothesis. — The attempt must now be made to derive some theo- 

 retical conclusions from the experiments detailed in the above paragraphs. As 

 before let n be the number of particles or ions per cubic centim., so that n is the 

 concentration or density of distribution of the [thosphorus emanation. Let k be 

 the "absorption" velocity of the ion as defined in Chapter III., treated in the first 

 instance as independent of the potential, and of the concentration gradients. Let 

 h' be the coefficient of decay, so that h'rv' is the number of ions vanishing per cubic 

 centim., per second. Finally, let B be the extei-nal radius of the condenser, and 

 its effective capacity including that of the electrometer. 



With regard to the electrical currents, let V be the potential at a distance r 

 from the center of the condenser whose external face is put to earth. Let U be 

 the ao-o-recate velocity of the ions in the unit electric field and e the chaige of each. 

 In all cases the observations are made when the flux is stationaiy, so that 

 dn/dt — 0, throughout for any shell. Moreover as shown above, the effect of a 

 potential gradient is but a negligible contribution to the number of ions which are 

 absorbed by the outer surface of the condenser. 



To begin with the simplest cases, if the motion of the ion is entirely independent 

 Of d V/dr and n, the accuratdation in an elementaiy shell at a distance r from the cen- 

 ter will be 4 71 hd (;rn)/dr. dr, per second ; the decay i>er second, h'n~ 4 n r dr. Hence 

 d(^i^n)/dr = (kf/]^/rcr, or, if A is a constant, \/n=r{(h' /h)+Ar). ^ In the absence 

 of decay, l/A — nr, so that A is the reciprocal of the concentration re,, at a dis- 

 tance 1 from the center. If conduction were pi'omoted solely by the ions which 

 reach the external shell kei)t at V= 0, since the charge in this shell is per square 

 centim., edM/[E(k'/k + AR)], and its time of discharge dli/k, Cd V/df = 

 in keli/il'/k + AJi), In the absence of decay, h' = 0, and dV/di = 4:7tkenJC, 

 where «,, as stated, holds for r = 1. This case in which d V/dt = & s/6 t = const, 

 independent of the radius of the condenser, is effectually excluded by the obsei'va- 

 tions as given in figure 11. If, however, h deci-eases with concentration, for which 

 there is some evidence in Chapter III., the case is still open. 



If h' is not zero, d V/dt =(4= nJce/C)ll/(k'/h E + ^)], so that the current 

 'Fis potential difference equivalent to s ; A and a are constants. 



