78 BELL SYSTEM TECHNICAL JOURNAL 



applied to any number of consecutive radioactive substances; there 

 are always just equations enough to determine all the constants and 

 describe completely the future history of any mixture of the members 

 of a single family line, provided that their relative proportions in the 

 mixture are specified for some particular moment. Even with only 

 three substances the behavior of the mixture may be extraordinarily 

 complicated; but there are simpler cases which are instructive. 



If for instance one sets aside a substance with a much longer half- 

 period than any of its posterity possesses, the extant quantity of each 

 and every one of the descendants will first increase and then begin to 

 decrease, and eventually diminish along the same exponential curve as 

 the long-lived ancestor itself — not because the half-periods of the 

 descendants are actually changed, but because of the partial balancing 

 between the decay and the replenishment of each. Thus the half- 

 period of the long-lived ancestor may be determined by plotting against 

 time the total intensity of all its rays and all the rays of its descendants, 

 or that of any particularly convenient kind of ray emitted by any 

 member of the family. The most carefully measured and accurately 

 known of all disintegration-constants, that of radon, is usually de- 

 termined in this way; its half-period amounts to four days, those of its 

 three next descendants radium A and radium B and radium C to only 

 a few minutes each, so that after isolating a sample of radon and 

 waiting a few hours one can set up any device for measuring the gamma- 

 rays of radium C, plot their decay-curve, and from it determine a 

 value of X which is not that of radium C, but that of radon. 



If in such a case as the foregoing the long-lived ancestor is so very 

 long-lived that no appreciable decrease in its rate of transmutation can 

 be detected over a period of years, then eventually the quantities and 

 the radiations of all of its descendants assume values which likewise do 

 not change appreciably for years ; ' ' radioactive equilibrium " is attained . 

 In a unit of time, equal numbers of atoms are transmuted out of each 

 substance into the substance following, into each substance out of the 

 one preceding. Representing by Nn the number of atoms of the «th 

 member of the series (counting the very long-lived ancestor as the 

 first) extant in the mixture in radioactive equilibrium, by X„ its 

 disintegration-constant, and remembering that \nNn is the rate at 

 which its atoms perish by transmutation, we have the chain of 

 equations: 



- dNSt = AiA^i = X2iV2 = \zN-i = XiNi = . . . (8) 



from which, if we know the relative quantities of any two members in 



