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~ Al *. 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 



Thus, = ^P - 1 2 Q = ,U e-M - Jl,Q. 



at 



The solution of this equation is of the form Q = oe~ Al * + be~^. 

 Since for a very short exposure Q = 



3 J 



AI AI 



and (/ == "i r~ ( c 6 ) , 



/!-X 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~^ i e~^ t . 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 ^! and ^ corresponding to the two changes. It is seen that the equa- 

 tion is symmetrical in ^ and X 2 and in consequence is unaltered if the 

 values of A t and ^ z are interchanged. Now the constant of the change 

 is determined by the observation that the activity finally decays to 

 half value in 1 1 hours. The theoretical and experimental curves are 

 found to coincide if one of the two products is half transformed in 1 1 

 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 1 1 hours. 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. 



