CASE 2. The activity curve for a long exposure to the emanation will now be considered. The activity after removal of _A_ and _C_ is proportional to λ₁_P_ + λ₃_R_, where the values of _P_ and _R_ are graphically shown in Fig. 75 by the curves _AA_, _CC_. Initially after removal, λ₁_P₀_ = λ₃_R₀_, since _A_ and _C_ are in radio-active equilibrium, and the same number of particles of each product break up per second. The activity due to _A_ alone is shown in curve _AA_, Fig. 75. The activity decreases exponentially, falling to half value in 3 minutes. The activity due to _C_ at any time is proportional to _R_, and is initially equal to that of _A_. The activity curve due to _C_ is thus represented by the curve _CC_, which is the same curve as the upper curve _CC_ of Fig. 73. The activity of _A_ and _C_ together is represented by the upper curve _A_ + _C_ (Fig. 75), where the ordinates are equal to the sum of the ordinates of the curves _A_ and _C_. This theoretical curve is seen to be very similar in shape to the experimental curve (Fig. 67) showing the decay of activity of the active deposit from a long exposure measured by the α rays. [Illustration: Fig. 75.] =203. Effect of a rayless change on the activity curves.= Certain important cases occur in the analysis of radio-active changes, when one of the products does not give rise to rays and so cannot be detected directly. The presence of this rayless change can, however, be readily observed by the variations which occur in the activity of the succeeding product. Let us consider, for example, the case where the inactive matter _A_, initially all of one kind, changes into the matter _B_ which gives out rays. The inactive matter _A_ is supposed to be transformed according to the same law as the radio-active products. Let λ₁, λ₂ be the constants of the change of _A_ and _B_ respectively. If _n_ is the number of particles of _A_, initially present, we see from the equation (4), section 197, that the number of particles of the matter _B_ present at any time is given by $$ Q = \frac {nλ_1} {λ_1 − λ_2} (e^{–λ_2 t} - e^{λ_1 t}) $$ . Differentiating and equating to zero, it is seen that the value of _Q_ passes through a maximum at a time _T_ given by the equation $$ λ_2 e^{–λ_2 T} = λ_1 e^{–λ_1 T} $$ . For the sake of illustration, we shall consider the variation of the activity of the active deposit of thorium, due to a very short exposure to the emanation. Thorium _A_ gives out no rays, and thorium _B_ gives out α, β, and γ rays, while thorium _C_ is inactive. The matter _A_ is half transformed in 11 hours, and _B_ is half transformed in 55 minutes. The value of λ₁ = 1·75 x 10⁻⁵(sec.)⁻¹ and λ₂ = 2·08 x 10⁻⁴(sec.)⁻¹. The activity of the mixture of products _A_ + _B_ is due to _B_ alone, and will, in consequence, be always proportional to the amount of _B_ present, that is, to the value of _Q_. [Illustration: Fig. 76.]