Voi,. 7, 1921 
BIOLOGY: A. J. LOTKA 
169 
It is not intended to enter here into this perfectly general case. It will suf- 
fice to point to the purely mathematical literature on the subject. 3 
2. Changes in the parameters A, P, Q proceeding very slowly as com- 
pared with the speed of readjustment of the X's. In this case the system, 
after passing through a "transient" state, ultimately settles down to a 
"moving equilibrium." The study of this case therefore comprises two 
phases, which may with advantage be taken up separately. The phase of 
the moving equilibrium is of particular interest, since such moving equili- 
bria play an important role in the evolution of physical systems, as pointed 
out years ago by Herbert Spencer. 4 
3. We may study the effect of a change in a parameter A, P or Q upon 
the equilibrium of the system alone, irrespective of the process by which 
that equilibrium is reached. It is in this case immaterial whether the 
change is slow or rapid. Into this division of the subject fall such relation 
as the principle of Le Chatelier, the thermodynamic laws of equilibrium 
and the "reciprocal relations" of statistical mechanics. 
The application of some of these principles to biological and social sys- 
tems has been essayed, but it cannot be said that the rigor of the attempts 
thus made is satisfactory. It would therefore be desirable to go over the 
ground and consolidate it. An effort in this direction is taken in view as 
part of the plan into which the present contribution is fitted. 5 
Of the general field outlined above, the portion to which we shall now 
give our attention is that of moving equilibria. 
Our fundamental system of equations we shall, for our present purpose, 
write in the form (2). Furthermore, merely in order to simplify our nota- 
tion, we will restrict the number of dependent variables to two, which, to 
avoid subscripts, we will denote by X, Y. We have, then 
dX/dt=Fi(X, F, t), dY/dt=F 2 (X, Y, t). (3) 
We adopt a method of successive approximations. Since the system is 
near equilibrium, we write for our first approximation 
0=F 1 (X,Y,t), 0=F 2 (X,y,*). (4) 
Solving for X and Y we then have 
Xx =<pi(t), Xx =Ht). (5) 
whence by differentiation 
dXi/dt=Xi' =Vi(2), dY 1 /dt = Y\ =^'i(0. (6) 
Proceeding to a second approximation, we substitute (6) in (3). 
<t>'i(t)=F h ♦'iW-ft. (7) 
Solving again for X, Y, 
X 2 = <P2(i), (8) 
We may again differentiate, 
XWi(*). IW'iG). (9) 
and, substituting as before, we obtain a third approximation. We con- 
tinue this process as far as may be desired, and finally obtain for the n th 
