Vol. 6, 1920 
BIOLOGY: A. J. LOTKA 
413 
We shall return later to the condition (11). 
Condition (12) we will employ to define a new origin. Accordingly we 
introduce into (8), (10) the variables: 
Bj 
A, 
A, 
= X, 
(13) 
(14) 
and obtain 
dxi 
dt 
dx2 
'di 
a^iXi + a2i2X\X2 
(15) 
where 
012 
am 
Oil 
(16) 
(17) 
(18) 
(19) 
_ BiB2 
A2 
A1A2 
B, 
0,2\2 — A2 
Note the significant fact that in (15) the linear terms in the dexter 
diagonal are lacking. It is this circumstance which imparts an oscillatory 
character to the process. 
For, since avi and 021 are in general functions of X\, X2, let us expand 
them by Taylor's theorem and put 
ai2 = + p\oo\ + piXi + (20) 
a2\ = + q\OC\ + qiX2 + (21) 
A general solution of the system of differential equations (15) is then 
= P^i-^' + P^e""'' + Pne''^^ + ^22^'"^'^ + . . . . (23) 
%i = Qie"-'' + 026^"-' + Qii^'"^' + Q22e'^'' + . . . . (24) 
where Xi, X2, are the roots of the determinental equation for X 
-X po 
Qo —X 
that is to say, 
= 0 
(25) 
^= ^^^PoQo , (26) 
Now, according to (20), (21) po, Qo are the equilibrium values of a^, 
a2i. Hence, if we denote by Ai, B2 the equihbrium values of Ai, B2, i.e., 
those values which correspond to Xi = X2 = 0, then we have, by (16), 
(18) 
^'»go = -A1B2 (27) 
and hence 
X = =bV=^2 (28) 
