it can readily be shown that
P = (n_{0}/λ_{1})e^{-λ_{1}t} (7),
Q = (n_{0}/(λ_{1} - λ_{2}))((λ_{1}/λ_{2})e^{-λ_{2}t} - e^{-λ_{1}t}) (8),
R = n_{0}(ae^{-λ_{1}t} + be^{-λ_{2}t} + ce^{-λ_{3}t}) (9),
where
a = λ_{2}/((λ_{1} - λ_{2})(λ_{1} - λ_{3})), b = -λ_{1}/((λ_{1} - λ_{2})(λ_{2} - λ_{3})),
c = λ_{1}λ_{2}/(λ_{3}(λ_{1} - λ_{3})(λ_{2} - λ_{3})).
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Relative number of atoms of matter A.B.C. present at any instant (Case 2).
Fig. 73.
The relative numbers of atoms of P, Q, R existing at any time are shown graphically in Fig. 73, curves A, B, C respectively. The number of atoms R_{0} is taken as 100 for comparison, and the values of λ_{1}, λ_{2}, λ_{3} are taken corresponding to the 3, 21, and 28-minute changes in the active deposit of radium. A comparison with Fig. 72 for a short exposure brings out very clearly the variation in the relative amounts of P, Q, R in the two cases. Initially the amount of R decreases very slowly. This is a result of the fact that the supply of C due to the breaking up of B at
