394 Reports on Specl\l Researches 



The method of calculating, from the decay curve, the number of atoms of radium A, 

 radium B, and radium C in the atmosphere, is analogous to that adopted in the case of the 

 observations of the third cruise of the Carnegie} The following notation will be adopted: 



Let \a, ^b, ^c, be the decay constants for the A, B, and C products of radium emana- 

 tion respectively; T the time of exposure of the foil to the atmosphere. 



Let Ha. be the number of atoms of radium A deposited per second during period T. 



Let riB be the number of atoms of radium B deposited per second during period T. 



Let ric be the number of atoms of radium C deposited per second during period T. 



Let t be the time at any instant after the foil ceases to be exposed to the atmosphere, 

 the time t = being the instant when the exposed foil is discharged. 



Let Na be the number of atoms of radium A on foil at time t due to direct deposition. 



Let Nb be the number of atoms of radium B on foil at time t due to direct deposition. 



Let Nc be the number of atoms of radium C on foil at the time < due to direct deposition. 



Let Nb be the number of atoms of radium B on foil at time t due to their formation 

 on the foil from radium A. 



Let Nc be the number of atoms of radium C on foil at time t due to their formation 

 on the foil from radium A via radium B. 



Let Nc be the number of atoms of radium C on foil at time t due to formation from 

 the radium B deposited. 



In the present work, T is 1,800 seconds, and the following values are adopted for 

 X^, Xb, and Xc: 



X^ = 3.85X10-^ \b = 4.33X10-" Xc = 5.93X10-* 



Proceeding on lines analogous to those indicated in Rutherford's "Radioactive Sub- 

 stances and their Transformations," pages 426-427, we obtain, 



\A^A = nAl-e->^A'')e-^Ai (8) 



XciVc = n4Xc(ae-M+be-^fi'+ce-M) (9) 



where 



X^(l-e-x^r) X^d-e-^^r) X>B(l-e-^cr) 



(Xb-XJ(Xc-X^) (X.4-X5)(Xc-Xb) ^ Xc(X^-Xc)(Xs-Xc) 



a = 

 We also find 



where 



XBiVfl = nB(l-e-^B^)e-M (10) 



\cNc = nB{hfi-HT-cfi-^d) (11) 



^ _ \c{l-e-^BT) ^ _ XB(l-e-^cr) 



^ Xp— Xs y^c~^B 



\cNc = nc(X-e-^cT)e-^c^ (12) 



If p, q, and r represent the number of ions produced by an a particle of radium A, 

 radium B, and radium C respectively, then, remembering that radium B emits no a-particles, 

 the total rate of production of ions which would take place in the ionization chamber as a 

 result of the deposit on the foil, if all the a-particles produced their full number of ions, is 



'\aNaV +^cNc'r +\c^(fr+\cNcr 



'See W. F. G. Swann, Terr. Mag-, vol. 20. pp. 30-43, 1915. 



