182 PROFESSOR E. RUTHEKFOKlt ON IIIK 



The solution of this equation is of the form 



Q = n (ae~* + be-*) . . . 

 By substitution it is found that a = X^X, X t ). 



Since Q = when t = 0, b = X t (X. 2 X t ). 

 Thus 



A i ^ A,-) 



(4). 



Substituting this value of Q in (2), it can readily be shown that 



R = n (oe- A ' ( + be-* + ce~*) 



(5), 



where a = 



X,X, 



(A,'- A,) (X, - X 3 ) ' 



1 = 



_ ~ *i*d 



( Xl _ X 2 ) (x, _ A 3 ) ' 



^.-y^-Xs)' 



The variation of the values of P, Q, R with the time t after removal is shown 

 graphically in n'g. 8, curves A, B, and C respectively. In order to draw the curves 



IOO 



30 



45 60 



Time in Minutes. 



Fig. 8. 



90 



105 



for the practical case corresponding to the first three changes in radium A, the values 

 of A,, Ag, X 3 were taken as 3'85 X 10~ s , 5'38 X 10~ + , 4'13 X 1CT* respectively, i.e., the 

 times required for each type of matter to be half transformed are about 3, 21, and 

 28 minutes respectively. 



The ordinates of the curves represent the relative number of atoms of the matter 

 A, B, and existing at any time, the value of n, the original number of atoms of the 



