80 BELL SYSTEM TECHNICAL JOURNAL 



One of the two best methods for determining the half-period of 

 radium is a combination of this last-named method with one of those 

 which I described earlier. Let us suppose that a sample of ionium, 

 equal to the amount which would be in equilibrium with one gramme 

 of uranium and 3.40- 10~^ grammes of radium, is purified of its original 

 radium-content and set aside for occasional observations of the rate of 

 growth of fresh radium in it. Representing by Ni the number of 

 ionium atoms in the sample (which diminishes in so small a proportion 

 that we may consider it constant), by A''2 the number of radium atoms 

 extant at time / after the new supply begins to grow, by Xi and X2 the 

 disintegration-constants of these two substances; translating equation 

 (4) into this notation, and remembering that the rate of transmutation 

 of the parent substance which was there called B is now (measured in 

 atoms transmuted per second) equal to \\Ni, we have 



iV2=^(l-.-^^0. (9) 



A2 



Represent by N^o the number of atoms of radium which would be in 

 radioactive equilibrium with the sample of ionium, that is to say, the 

 number of atoms in 3.40- lO^'^ grammes of radium; by equation (8) we 

 have 



XiA^i = X2A^2o, (10) 



so that equation (9) may be transformed into one containing no 

 constants except the known one iV2o and the object X2 of the investi- 

 gation. The gain is still greater; developing the exponential function 

 in (9) as a power-series in t and retaining only the first term, we have 



iVo = A^2o(l - e"^'0 = ^^20X2/ -f terms of higher order. (11) 



This means that we need to trace the growth-curve of radium out of 

 ionium only so far as is necessary to determine its initial slope, the 

 initial rate at which the radium increases before its own transmutation 

 begins to tell. This as it happens is all that there has yet been time to 

 trace, so that this combination of the two methods is the only way yet 

 available of interpreting the growth-curves.^" After a sample of 

 ionium has been kept for a century or two, it may be possible to trace a 

 long enough arc of the curve to determine by the first method. After 



ascribed to the comparative youth of the rocks, in others to the selective action of 

 flowing water and other geological agents in removing some and leaving others of the 

 members of the radioactive family. 



-° This method, it will be perceived, is essentially a measurement of one and hence 

 of all of the terms X„iV„ which are equated in equation (9); the rate of growth of 

 radium out of ionium being ascertained, it is possible to calculate the value of Xn 

 for any substance in the radium series for which Nn, the quantity in equilibrium with 

 the preassigned quantity of ionium, can be measured. 



