SUCCESSION OF CHANGES IN RADIOACTIVE BODIES. 181 



(2) A second ' rayless ' change in which half the matter is transformed in 



21 minutes ; 



(3) A third change in which half the matter is transformed in 28 minutes, and 



which gives rise to a, ft, and y raya 



y. Theory of Successive Changes. Before considering the evidence from which 

 these changes are deduced, the general theory of successive changes of* radioactive 

 matter will be considered. It is supposed that the matter A deposited by the 

 emanation changes into B, B into C, C into D, and so on. 



Each of these changes is supposed to take place according to the same law as a 

 monomolecular change in chemistry, i.e., the number N of particles unchanged after a 

 time t is given by N = N e~ w , where N is the initial number and X the constant of 

 the change. 



Since dN/dt = XN, the rate of change at any time is always proportional to the 

 amount of matter unchanged. It has previously been pointed out that this law ot 

 decay of the activity of the radioactive products is an expression of the fact that the 

 change is of the same type as a monomolecular chemical change. 



Suppose that P, Q, R represent the number of particles of the matter A, B, and C 

 respectively at any time t. Let X,, Xj, X 3 be the constants of change of the matter 

 A, B, and C respectively. 



Each atom of the matter A is supposed to give rise to one atom of the matter B, 

 one atom of B to one of C, and so on. 



The expelled ' rays ' or particles are non-radioactive and so do not enter into the 

 theory. 



The general theory will first be considered corresponding to the practical cases of a 

 very short and of a very long exposure in the presence of the emanation, then finally 

 for any time of exposure to a constant supply of the emanation. 



10. Short Exposure. Suppose that a body has been exposed for a short interval 

 in the presence of the radium or thorium emanation and then removed. The time is 

 supposed to t)e so short that no appreciable portion of the deposited matter has 

 undergone change during the time of exposure. It is required to find the number of 

 particles P, Q, R of the matter A, B, C respectively present after any time t. 



Then P = ne~ X|( , if n is the number of particles that has been deposited during the 

 short time of exposure. Now the increase of the number of particles rfQ of the 

 matter B per unit time is the number supplied by the change in the matter A, less 

 the number due to the change of B into C, thus 



dQ/^ = X t P-X 2 Q (1). 



Similarly 



dR/dt = XjjQ - XjR (2). 



in (1) the value of P in terms of n, 



