CONTEMPORARY ADVANCES IN PHYSICS 77 



of X is determined from it quite easily, and as a matter of fact the 

 decay-curve itself can be obtained merely by plotting as function of 

 time the difference between Q and the limiting-value (= B/\) which 

 Q approaches as t increases indefinitely. Determining a half-period 

 from a rate of growth is therefore mathematically the same process as 

 determining it from a rate of decay. This is one of the ways in which 

 the half-period of uranium Xi is measured; and the standard method 

 for determining that of radium is based partly upon it, as we shall 

 presently see. 



Eventually the grandchild and the remoter posterity of the parent- 

 substance must make their presence known. This is not always a 

 disadvantage. Letting Xi and Qi stand for the disintegration-constant 

 and the extant quantity of the daughter-substance, X2 and Q2 for those 

 of the granddaughter, we have as basis for the theory these equations: 



dQ,/dt = B - \,Qu dQ^/dt = Xi()i - X.^, (5) 



integrating which, and supposing that at / = the parent-substance 

 has just been isolated so that the building-up of the two descendants 

 from zero is just commencing, we obtain for Qi the expression (4) with 

 Xi in the place of X, and for Qo the function 



Q^ = ^\ —J t ^-Xi< ^] ^-X2< 



L X2 ^ Xi - 



Xo XsCXi - X2) 



which to second approximation is equivalent to 



(6) 



Q2=\B\if. (7) 



The amount of the grandchild therefore should increase at first as the 

 square of the time elapsed, whereas the amount of the child increases 

 proportionally to the time. There are instances, in the history of the 

 study of radioactivity, of a substance being regarded as the child of 

 another until measurements were made upon its rate of growth in an 

 isolated sample of its putative parent, whereupon through its con- 

 formity to (7) it was proved to be the grandchild and not the child. 

 The question whether radium comes directly out of uranium II, or out 

 of an intermediate substance, was settled in this fashion; and by 

 observing a sample of uranium II at intervals over a period of almost 

 twenty years, and measuring the radium which was being developed 

 within it, Soddy was able through equation (7) to calculate the half- 

 period of this intermediate substance (ionium). 



The method used in deriving the equations (4) and (7) can always be 



