appear, and these are quite distinct in chemical and physical
properties from the parent matter. The radiating power is an
atomic property, for it is unaffected by combination of the
active element with inactive bodies, and is uninfluenced by the
most powerful chemical and physical agencies at our command.
In order to explain these results, Rutherford and Soddy (20)
in 1903 put forward a simple but comprehensive theory. The
atoms of radioactive matter are unstable, and each second a
definite fraction of the number of atoms present break up with
explosive violence, in most cases expelling an α or β particle
with great velocity. Taking as a simple illustration that an
α particle is expelled during the explosion, the resulting atom
has decreased in mass and possesses chemical and physical
properties entirely distinct from the parent atom. A new type
of matter has thus appeared as a result of the transformation.
The atoms of this new matter are again unstable and break up
in turn, the process of successive disintegration of the atom
continuing through a number of distinct stages. On this view,
a substance like the radium emanation is derived from the
transformation of radium. The atoms of the emanation are
far more unstable than the atoms of radium, and break up at
a much quicker rate. We shall now consider the law of radioactive
transformation according to this theory. It is experimentally
observed that in all simple radioactive substances,
the tensity of the radiation decreases in a geometrical progression
with the time, *i.e.* I/I_{0}=e^{-λt} where I is the intensity
of the radiation at any time t, I_{0} the initial intensity, and λ
a constant. Now according to this theory, the intensity of the
radiation is proportional to the number of atoms breaking up
per second. From this it follows that the atoms of active
matter present decrease in a geometrical progression with the
time, *i.e.* N/N_{0}=e^{-λt} where N is the number of atoms present
at a time t, N_{0} the initial number, and λ the same constant
as before. Differentiating, we have dN/dt=-λN, *i.e.* λ represents
the fraction of the total number of atoms present which
break up per second. The radioactive constant λ has a definite
and characteristic value for each type of matter. Since λ is
usually a very small fraction, it is convenient to distinguish
the products by stating the time required for half the matter
to be transformed. This will be called the period of the product,
and is numerically equal to log Qe^{2}/λ. As far as our observation
has gone, the law of radioactive change is applicable to all
radioactive matter without exception. It appears to be an
expression of the law of probability, for the average number
breaking up per second is proportional to the number present.
Viewed from this point of view, the number of atoms breaking
up per second should have a certain average value, but the
number from second to second should vary within certain
limits according to the theory of probability. The theory of
this effect was first put forward by Schweidler, and has since
been verified by a number of experimenters, including Kohlrausch,
Meyer, and Begener and H. Geiger. This variation
in the number of atoms breaking up from moment to moment
becomes marked with weak radioactive matter, where only a
few atoms break up per second. The variations observed are
in good agreement with those to be expected from the theory
of probability. This effect does not in any way invalidate the
law of radioactive change. On an average the number of
atoms of any simple kind of matter breaking up per second is
proportional to the number present. We shall now consider
how the amount of radioactive matter which is supplied at a
constant rate from a source varies with the time. For clearness,
we shall take the case of the production of emanation, by
radium. The rate of transformation of radium is so slow
compared with that of the emanation that we may assume
without sensible error that the number of atoms of radium
breaking up per second, *i.e.* the supply of fresh emanation, is
on the average constant over the interval required. Suppose
that initially radium is completely freed from emanation. In
consequence of the steady supply, the amount of emanation
present increases, but not at a constant rate, for the emanation is
in turn breaking up. Let q be the number of atoms of emanation
produced by the radium per second and N the number present
after an interval t, then dN/dt=q-λN where λ is the radioactive
constant of the emanation. It is obvious that a steady
state will ultimately be reached when the number of atoms
of emanation supplied per second are on the average to the
atoms which break up per second. If N_{0} be the maximum
number, q=λN_{0}. Integrating the above equation, it follows
that N/N_{0}=1-e^{-λt}. If a curve be plotted with N as
ordinates and time as abscissae, it is seen that the recovery
curve is complementary to the decay curve. The two curves
for the radium emanation period, 3.9 days, are shown in fig. 1,
the maximum ordinate being in each case 1oo.

This process of production and disappearance of active matter holds for all the radioactive bodies. We shall now consider some special cases of the variation of the amount of active matter with time which have proved of great importance in the analysis of radioactive changes.

(*a*) Suppose that initially the matter A is present, and this changes
into B and B into C, it is required to find the number of atoms P, Q
and R of A, B and C present at any subsequent time t. .

Let λ_{1}, λ_{2}, λ_{3} be the constants of transformation of A, B and C
respectively. Suppose n be the number of atoms of A initially
present. From the law of radioactive change it follows:

P=ne^{-λ1t}

(1)

dQ/dt=λ_{1}P-λ_{2}Q

(2)

dR/dt=λ_{2}Q-λ_{3}R

Substituting the value of P in terms of n in (1), dQ/dt = λ_{1}ne^{-λ1t}-λ_{2}Q;
the solution of which is of the form

Q=n(ae^{-λ1t}+be^{-λ2t}),

where a and b are constants. By substitution it is seen that
a=λ_{1}/(λ_{2}-λ_{1}). Since Q=0 when t=0, b= -λ_{1}/(λ_{2}-λ_{1})

(3)Thus Q = nλ_{1}λ_{1}-λ_{2}(e^{-λ2t}-e^{-λ1t})

Similarly it can be shown that

(4)

R=n(ae^{-λ1t}+be^{-λ2t}+ce^{-λ3t})

where a=λ_{1}λ_{2}(λ_{1}-λ_{2})(λ_{1}-λ_{3}), b=λ_{1}λ_{2}(λ_{2}-λ_{1})(λ_{2}-λ_{3}), c=λ_{1}λ_{2}(λ_{3}-λ_{1})(λ_{3}-λ_{2})

It will be seen from (3), that the value of Q, initially zero, increases to a maximum and then decays; finally, according to an exponential law, with the period of the more slowly transformed product, whether A or B.

(*b*) A primary source supplies the matter A at a constant rate,
and the process has continued so long that the amounts of the
products A, B, C have reached a steady limiting value. The primary
source is then suddenly removed. It is required to find the amounts
of A, B and C remaining at any subsequent time t.

In this case of equilibrium, the number n_{0} of particles of A
supplied per second from the source is equal to the number of particles
which change into B per second, and also of B into C. This requires
the relation

n_{0}=λ_{1}P_{0}=y_{2}Q_{0}=λ_{3}R_{0}

where P_{0}, Q_{0}, R_{0} are the initial number of particles of A, B, C
present, and λ_{1}, λ_{2}, λ_{3} are their constants of transformation.
Using the same quotations as in case (1), but remembering the
new initial conditions, it can easily be shown that the number of
particles P, Q and R of the matter A, B and C existing at the time
I after removal are given by

P=n_{0}λ_{1}e^{-λ1t},

Q=n_{0}λ_{1}-λ_{2}λ_{1}λ_{2}e^{-λ2t}-e^{-λ1t},

R=n_{0}(ae^{-λ1t}+be^{-λ2t}+ce^{-λ3t}),

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})

The curves expressing the rate of variation of P, Q, R with time
are in these cases very different from case (1).
(*c*) The matter A is supplied at a constant rate from a primary
source. Required to find the number of particles of A, B and C
present at any time t later, when initially A, B, and C were
absent.

This is a converse case from case (2) and the solutions can be
obtained from general considerations. Initially suppose A, B and
C are in equilibrium with the primary source which supplied A at a
constant rate. The source is then removed and the amounts of
A, B and C vary according to the equation given in case (2). The
source after removal continues to supply A at the same rate as
before. Since initially the product A was in equilibrium with the
source, and the radioactive processes are in no way changed by the
removal of the source, it is clear that the amount of A present in
the two parts in which the matter is distributed is unchanged. If
P_{1}, be the amount of A produced by the source in the time t, and P