PRESENT PROBLEMS OF RADIOACTIVITY 177 



of which alone gives out rays. The matter deposited on the body 

 during the short exposure consists almost entirely of thorium A. 

 Thorium A changes into B and the breaking up of B gives rise to 

 the activity measured. 



Let n = number of particles of thorium A deposited on the body 

 during the time of exposure to the emanation. 



Let P and Q be the number of particles of thorium A and B re- 

 spectively at any time after removal. 



Let ^i, ^2 be the constants of the two changes. 



The number of particles of P existing at any time t is given by 

 p = n e~^ lt . If each atom of A in breaking up gives rise to one atom 

 of B, the increase dQ in the number of Q in the time dt is given by 

 the difference between the number of atoms of B supplied by the 

 change in A and the number of B which break up. 



dQ_ 



-L I 1 I I f 1 j I ' ^ * f^ "2^t ' /\ I / 6n/ **2 M 



dt 



The solution of this equation is of the form Q = ae~^ + be~^. 

 Since for a very short exposure Q = 



Q, = = T 



and Q = --(c-^-e-^}. 



AI /2 



If thorium A does not give out rays, the activity of the body at any 

 time after removal is proportional to Q. Thus the activity at any 

 time t is proportional to e~^ e~^. Now the experimental curve 

 of variation of activity is found to be accurately expressed by an equa- 

 tion of this form. A very interesting point arises in settling the values 

 of A! and ^ corresponding to the two changes. It is seen that the equa- 

 tion is symmetrical in h and h and in consequence is unaltered if the 

 values of ^ and X 2 are interchanged. Now the constant of the change 

 is determined by the observation that the activity finally decays to 

 half value in 11 hours. The theoretical and experimental curves are 

 found to coincide if one of the two products is half transformed in 11 

 hours and the other in 55 minutes. The comparison of the theoretical 

 and experimental curves does not, however, allow us to settle whether 

 the period of change of thorium A is 55 minutes or 11 hour,s. In order 

 to settle the point, it is necessary to find some means of separating 

 the products thorium A and B from each other. In the case of tho- 

 rium, this is done by electrolysing a solution of thorium. Pegram 

 obtained an active product which decayed according to an exponen- 

 tial law with the time falling to half value in a little less than 1 hour. 

 This result shows that the radiating product thorium B has the 

 shorter period. In a similar way, by recourse to electrolysis, it has 

 been found that the change of actinium B has a period of 1.5 minutes. 



