occurring in the theory of radio-active transformations. 425 



In the case when Q (0), R (0), . . . are not zero, we have 



^ « + Xi' ^ {so + Xi) (x + Xi) oc + Xo 



X.1X2-P0 , X^^^Q 



r = 



+ 



+ 



Hn 



...(8), 



{oc-{-Xi){oG + X2)(sc + Xa) {x + Xz){x + \s) (os + Xs) 

 etc. / 



and we may obtain the values of P, Q, R by expressing these 

 quantities in partial fractions as before. 



The complete solution for the case of a primary substance P 

 and three products Q, R, S is 



R = 



S = 



A-j A,2 -to 



(Xg — Xi) (X-s — Xi) 



e-^if + 



XjXgXsX^O 



(X.3-Ai)(X3-Xi)(X4-\i) 



XiXgXs-To 



X1X2X0 Xg^/p 



_(Xi — X2) (Xg — X2) Xj — X. 

 X1X2P0 X2Q0 



L (^1 ~ ^3) (X2 — X3) X2 — X3 



+ Ro 



+ 

 + 



+ 

 + 



+ 



X2X3Q0 



g-A3f^ 



o-Kof 



{X, - X2) (X3 - X2) (X4 - X2) (X3 - X2) (X, - X2) J 



Xi X2 X3 -T X2 X3 (j^o 



_(Xi — X3) (Xg — X3) (X4 — X3) (X2 — X3) (X4 — X3) 



X^Rq 



Xa Xq 



Q-ht + 



Xi X2 X3 Pq 



XoXs^o 



(Xi - X4) (X2 - X4) (X3 - X4) 

 X3R0 



+ 



+ So 



-Kit 



(X2 — X4) (X3 — X4) X3 — X4 



The solution may evidently be obtained by superposing the 

 solutions of the cases in which the initial values of P, Q, R, S are 

 given by 



(1) P(0) = Po, Q{0) = 0, R{0) = 0, S{0) = 0. 



(2) P(0) = 0, Q(0)=Qo, R{0) = 0, S{0) = 0. 



(3) P(0) = 0, Q(0) = 0, R(0) = Ro, S{0) = 0. 



(4) P(0) = 0, Q(0) = 0, P(0) = 0, ;Sf(0) = ^o. 



The method is perfectly general, and the corresponding 

 formulae for the case o( n—1 products may be written down at 

 once by using (6). 



The general formula covers all the four cases {Radio-activity, 

 pp. 331 — 337). For instance in Case 2 when initially there is 

 radio-active equilibrium, Ave have 



Uf) = XjPo = XnQ^^ = XgPo = X^Oq, 



