170 
BIOLOGY: A. J. LOTKA 
Proc. N. A. S. 
approximation 
X n = <p n (t), Y n =Mi). (10) 
If the functions <p n , and <p' n , \l/' n , obtained by these successive approx- 
imations tend toward a limit as n is increased indefinitely, then it can 
easily be shown that (10) is a solution of (3). 
It seems to be somewhat difficult to give in general terms the conditions 
for this convergence of successive approximations toward a limit. How- 
ever, the following example will show how in individual cases the question 
regarding this convergence can sometimes be readily answered. 
For our example we take the case of radioactive equilibrium. We have 
a chain of transformations 
A—>B—>C—>D (11) 
We will denote the masses of A, B,C,D at time t, respectively, by U, X, Y y 
Z, and their values at time 2 = 0 by the same letters with the subscript 
zero. 
We may if we choose (this is a purely arbitrary matter) pick out the sub- 
stances B and C for our evolving system, and look upon A and D as exter- 
nal factors influencing the system. The system composed of B and C is 
then subject to an equation of constraint 
X + Y = X 0 +Yo+U 0 - U + Zo- Z (12) 
in which the quantities appearing in the right hand member are param- 
eters of the class denoted above in equation (1) by A, that is to say, par- 
ameters introduced by the equation of constraint. Four of them are con- 
stants, the other two are functions of t. Of these last two one, namely 
U y will appear in the equations representing the course of evolution of the 
system composed of B and C. These equations, according to the well- 
known laws of radioactive transformations are 
dX/dt = aU - bX, dY/dt = bX - cY. (13) 
where a, b, c are constants. U, on the other hand, is a function of the time, 
namely 
U=Uoe~ at (14) 
For our first approximation we put, then, 
dX/dt = 0 = aUoe- at -bX 1 dY/dt = 0 = bX 1 -cY 1 (15) 
whence 
Xi = a/b. U 0 e- at , Y^a/c. U«e- at , (16) 
and therefore 
X\ = -a*/b. U 0 e- at , Y\ = - a*/c. U&r at . (17) 
The second approximation now gives 
dX/dt=X\=-ayb. U 0 e- at ^aU^- at - bX 2 (18) 
dY/dt-Y'i=-a*/c. Uoe- at = bX 2 - cY 2 (19) 
Whence 
X 2 = a/b. U«e- at ( 1 + a/b) (20) 
Yz = a/c. Uoe- at ( 1 + a/b + a/c) (21) 
