384 JOURNAL OF THE WASHINGTON ACADEMY OF SCIENCES VOL. U, NO. 16 



quently abbreviated to the form "% mg. of radium-C," or to ''% milli- 

 curies of radium-C." Such abbreviations may lead to serious mis- 

 understanding and confusion, and can readily be replaced by the 

 exact and unambiguous expression "x mr of radium-C," so soon as a 

 name has been given to the quantity r. 



The customary expression for the rate of accumulation of a radio- 

 element is 





= Xl A^2 - >^2 ^2 



where A^i and N2 denote, respectively, the total number of atoms of 

 two successive elements. The rate of accumulation of element 2 is 

 made to depend upon both Xi and Xo, although it is evident that the 

 instantaneous growth of element 2 can ultimately depend upon only 

 its own constant and its departure from equilibrium. This is brought 

 out plainly when quantities of the elements are measured in terms of 

 r. Let there be Ri r's of element 1 and R2 r's of element 2 ; as before, 

 let k denote the number of transformed atoms produced in unit time 

 by one gram of radium; i.e., by one r. Then kR2 will be the number 

 of atoms of element 2 that transforms in a unit of time, hence kRo = 

 XiNt, where N2 is the number of atoms of element 2. Whence 



dA/"2 ^. dRa 



"d^ ^2 d^ 



But 



dt 

 Hence 



dRo 



dt 



— X2 [Ri — R2) 



If Ri = R2 the two elements are in equilibrium. The differential 

 equation shows at once that the instantaneous growth of element 

 2 is equal to the product of X., by the departure from equilibrium. 



The coefficients in the equation applying to the decay of a group of 

 elements initially in equilibrium (Case 2 of Rutherford) are unsym- 

 metrical when the equations are written in the usual notation, the 

 total number of atoms present being the unit. For example, the 

 coefficients for the third element (Rutherford's R) are 



X2 Xi Xi X2 



a = 7^ — TTTi^ — n; = 



(X2-Xi)(X3-Xi)' " ~ (Xi-X2)(X3-X2)' ' ~ X3(Xi-X3)(X2-X3)" 



