76 BELL SYSTEM TECHNICAL JOURNAL 



of a mixture of substances of which the initial composition is taken for 

 granted. There is no better way of conveying a notion of the methods 

 by which radioactivity was and is studied than to describe how some 

 of the known half-periods were actually ascertained. 



The simplest of all the cases are those in which a substance which can 

 easily be separated from its ancestors transmutes itself into one which 

 either is not radioactive at all, or else decays so slowly that the rays 

 which it emits are not strong enough to interfere with the observations 

 on the rays of its parent. The penultimate substances of the various 

 series are candidates for this class, but the only one among them which 

 is abundant enough and lasts long enough to be easily isolated from its 

 ancestors is radium F, otherwise known as polonium. This therefore 

 is the classical instance of a substance of which the decay-curve is 

 determined directly from observations on rays of its own. Another 

 is radium E, of which the half-period is so short (about 5 days) and 

 the half-period of its daughter-substance so long (more than four 

 months) that its decay-curve can be traced practically as if it changed 

 into a stable element (Fig. 2). 



Almost as simple are certain cases in which a radioactive substance 

 is isolated both from its ancestors and from its posterity, and then the 

 growth of its immediate descendant is measured. This method is 

 available when the parent-substance is much longer-lived than its 

 child, so that the rate at which atoms of the latter come into being is 

 practically constant throughout the period of observation. Let B 

 represent this rate; let Q represent the quantity of the daughter- 

 substance extant at any time /, the time being measured from the 

 instant when the isolation of the parent-substance is perfected, so that 

 ^ = at / = 0; let X stand for the disintegration-constant of the 

 daughter-substance, so that the rate at which its atoms are disap- 

 pearing through transmutation is equal to \Q. The net rate of growth 

 of the daughter-substance is therefore 



dQIdt = -\Q + B (3) 



from which we obtain by integration 



<2 = ^ (1 - e-'O, (4) 



so that the quantity of the daughter-substance, and the intensity of its 

 rays vary as exponential functions of time with the disintegration- 

 constant standing in the exponent. This function, it is true, rises 

 from zero to a positive final limiting-value instead of falling to zero 

 from a positive initial value, as the decay-curve would; but the value 



