Ratios of Substances in Radioactive Equilibrium. 353 



To many experimental physicists calculus is a thorn in the 

 flesh and a weariness to the spirit, partly no doubt because 

 they so constantly have to deal with functions whose exact 

 mathematical expression is unknown or uncertain. I take 

 leave to suggest that mechanical quadratures may serve them 

 to reach conclusions which mathematicians would obtain more 

 elegantly but in a less obvious way. 



But it appears to me that there is also room for a methodical 

 examination by mathematicians of the applicability of: 

 formulae such as are developed in this paper to the integra- 

 tion of functions which cannot be integrated in " finite 

 terms." In such an inquiry the main point would be to 

 determine for each class of functions the limits of convergence 

 of Euler's series and the nature of the substitutions most 

 conducive to increase in convergence. I hope somebody may 

 pursue the matter further. 



Washington, D.C. 

 April, 1911. 



XXXIV. On the Ratios- which the Amounts of Substances 

 in Radioactive Equilibrium bear to one anotlu 



ler. 



To the Editors of the Philosophical Magazine. 

 Gentlemen, — ■ 



ON page 40 of the last volume of this Magazine Mr. H. 

 Mitchell has given a general solution of the system of 

 differential equations representing the course of a series 

 of consecutive radioactive transformations. This solution is 

 then applied to the case in which the mean length of life 

 of the primary parent substance is greater than that of any 

 other member in the series. Mr. Mitchell's result is more 

 general than the one usually cited, which depends on the 

 assumption that the parent substance not only has the 

 smallest rate of decay, but that this rate is negligible as 

 compared with any other in the series. 



It appears worth noting that the solution given by 

 Mr. H. Mitchell is capable of still more general interpretation. 

 For if the ?nth substance is the longest lived, we find, by a 

 process precisely similar to that given in the communication 

 referred to, 



= — , when n > m, 



X m + X ™ + 1 + ®m + 2 . . . + . C n \ n 



and # n =0, when ?i<m. 



Phil Mag. S. 6. Vol. 22. No. 128. Aug. 1911. 2 A 



