Mr Bateman, Solution of a system of differential equations, etc. 423 



The solution of a system of diffh-ential equations occurring 

 in tJie theory of radio-active transformations. By H. Bateman, 

 M.A., Trinity College. 



[Read 21 February 1910.] 



1. It has been shown by Prof. Rutherford. * that the amounts 

 of the primary substance and the different products in a given 

 quantity of radio-active matter vary according to the system of 

 differential equations, 





XzQ 



~Tr — ~X^Q ~ XsSi 



d^ 

 dt 



= X-xR — \aT 



(1), 



where P, Q, R,S,T,... denote the number of atoms of the primary 

 substance and successive products which are present at time t. 



Prof Rutherford has worked out the various cases in which 

 there are only two products in addition to the primary substance, 

 and it looks at first sight as if the results may be extended to any 

 number of products without much labour. 



Unfortunately the straightforward method is unsymmetrical 

 and laborious, and as the results of the calculations are needed in 

 some of the researches which are being carried on in radio-activity 

 the author has thought it worth while to publish a simple and 

 symmetrical method of obtaining the required formulae. 



2. Let us introduce a set of auxiliary quantities p (x), q (x), 

 r{x), ... depending on a variable x and connected with the 

 quantities P {t), Q (t), R (t), ... by the equations, 



y. 00 -.00 



p{x)= e-'''P(t)dt, q(x)=\ e-''^Q{t)dt (2). 



It is easily seen that 

 dP 



f 



Jo 



dt 



-P{0) + x I e-^* P (t) dt 



JO 

 — Po + Xp, 



•(3), 



Radio-activity, 2nd edition, p. 331. 



