Substances in Radioactive Equilibrium bear to one another. 41 



this assumption the theoretically correct deduction from the 

 above law still remains very simple. 



(2) Let £ ;i be the number of atoms initially and x n the 

 number at time t of a radioactive substance S which changes 

 into a radioactive substance S, , n at the rate of X„ x atoms 

 per unit time, where n = 1, 2, 3 . . . n. Then the average life 



of an atom of £ 

 ential equation 



of an atom of S. is — units of time and x n satisfies the differ 



. n = X , v ^ — A. X . 



dt n ~ l n - 1 n n ' 



in which when n= 1, X ,r is to be taken as zero. 



(3) The general solution of this equation giving the 

 quantity x n of the nth substance after time t when the parent 

 substance is initially free from products is 



C e ~\,t € -\ 2 t € ~\ n t \ 



^-fi^-VDJnp^-xo + H(a^a7) + '"n(x i -x n )j 



where the expression II(X. — Xj is to be taken to mean the 

 product of all terms of the type (X. — XJ when A,- takes all 

 possible values from Ai to \ n except \ x . 

 For example, 



and 

 Vi = ?lXl ^ 3 ( (A,-X 1 )(A 3 6 -X 1 )(A 4 -X 1 ) + 



^i)(A 3 -x 1 )(x 4 -x 1 ) (^-x^-x^-x^ 



(X 1 -X 3 )(X 2 -X 3 )(A,-A 3 ) ' (X 1 -X 4 )(X 2 -X 4 )(A 3 -A 4 )J ' 

 (4) When the parent substance is the longest lived member 



+ ^ ^7. ^v^ ^v + 



of the series, as t increases, — approaches the value 



x x 



\{\o- ■ -A n _i 



(X 2 -X 1 ){X 3 -X 1 )...(X n -X 1 ) 

 N T ow 



i+ Al 4- — -hh + _ 



As~~Ai (X 2 — Xj) (X 3 — Ai) 



I X 1 A 2 y--A( M _l) >oX 3 ...X, ? 



(A 2 -X 1 )(X 3 -X 1 )....(An-X 1 ) (\ 2 -\i)(A 8 -Ai)...(A»- Ai)' 



