Vol. 7, 1921 
BIOLOGY: A. J. LOTKA 
.171 
Continuing this process we find, ultimately, 
X n = a/b U 0 e- at (1 + a/b + a 2 /b* +... ) (22) 
Y n = a/c U 0 e- at (1+ a/b + a/c + a 2 /6 2 + a 2 /be + a 2 /c* + . . .) (23) 
In this example the condition for the convergence of the successive ap- 
proximations is immediately apparent. We must have 
a/b<l, a/c<l; (24) 
that is to say, the parent substance A must have a smaller decay constant 
than any of the succeeding members of the series. For obvious reasons 
this condition is always satisfied in natural radioactive mixtures. 6 
The series (22), (23) bring out the relation between the uncorrected 
equilibrium, as commonly computed on the assumption of constancy of 
mass of the parent substance, and the true equilibrium. The first- 
mentioned (for which Rutherford has suggested the term "secular equi- 
librium") is represented by the first term of the series. As Rutherford 
points out, the error of the first approximation, i.e. the difference between 
the secular and the true equilibrium, amounts, in some cases, to nearly 1% 
though in others the error is quite negligible. 
The series are easily summed, and then lead to the well-known expres- 
sions obtained by other methods (for the equations of radioactive change 
are readily integrable in finite terms, while the method here developed is 
applicable also in more refractory cases). 
The case of radioactive equilibrium was here selected as an illustration, 
primarily because the functions involved are known and of simple form. 
But the same example will serve very aptly to illustrate also some other 
points. 
In the first place we observe that moving equilibria might be divided 
into three classes, according as their progress is determined by a change in 
the P's, the Q's or the A's. As has been shown, the radioactive equilib- 
rium is of the type in which the pace is set by a parameter of the class A, 
namely the mass of one of the links in the chain, which thus acts as a brake, 
or a limiting factor checking the series of transformations. Such limiting 
factors play an important r61e also in the highly complex network of inter- 
locking cycles upon which the continuance of abundant life upon the 
earth depends. For life processes are energy transformation processes 
carried out by the agency of material energy transformers. Such trans- 
formers, if they are to work continuously and indefinitely must perforce 
work in closed transformation chains or cycles (such as the cycle C0 2 — > 
Plant — > Animal — ^C0 2 ). The moving equilibria engendered in such sys- 
tems of cycles by a slow change in a limiting factor, in a parameter of 
class A, invite further study. The influence of man upon the world's 
events seems to have been largely to accelerate the circulation of matter 
and energy through such cycles, either by "enlarging the wheel", i.e., in- 
creasing the mass taking part in certain cycles, or else by causing it to 
