and its Products of Transformation. 833 



of radium F breaking up per second is A 4 S, which is equal to 



M 



The value of X 4 S is deduced experimentally from the counting 

 experiments* Since the values of X} and X 4 are known, the 

 value of X 2 , the constant of radium I), can at once be 

 deduced. 



In the more complete theory it is necessary to take into 

 account the period of the emanation and of radium E. The 

 emanation may be supposed to be transformed directly into 

 radium D, for the intervening products have short periods 

 of transformation. Let X 1? X 2 , X 3 , X 4 , be the constants of 

 transformation of the emanation, radium D, radium E, and 

 radium F respectively. Let N be the number of atoms of 

 emanation present initially, and S the number of atoms of 

 radium F present at time t after the introduction of the 

 emanation. Then 



The Values of the coefficients A, B, C, D were deduced for 

 me by Mr. H. Bateman, to whom I am very much indebted. 



. X, A 2 X 3 



(X 2 -X 1 )(X a -X 1 )f>4-^i) 



X]X 2 Xj 



c= 



(Xi — Xo)(X 3 — X 2 )(X 4 — X 2 ) 



A1A2A3 



^i-^)(^-X3)(X 4 -X 3 ) 



r)_ A1A2A 3 



a "(X 1 -X 4 )> 2 -X 4 )(X 3 -X 4 )' 



It can easily be shown that for the time interval in the 

 experiment, A e~ klt and Ge~* 3t are very small and can be 

 neglected. In order to obtain the values of B and D it is 

 necessary to know the constant of radium D approximately. 

 This can be done by the approximate theory already 

 mentioned. Writing the expression for B in the iorin 



^iL A2 jl_^2 i_^-J 



