414 
BIOLOGY: A. J. LOTKA 
Proc. N. a. 
Now the coefficient B2 is, in the nature of things, a positive number, as 
follows from its definition by (4), (9). 
As regards the coefficient Ai, we have two possible alternatives. 
If ^1 is negative for all values of X\, X2, then X, as defined by (28), 
would be real; but this inference is nugatory. F'or Bi, like is, by 
definition (3), (5), an essentially positive quantity, and hence the equilib- 
rium defined by (12) would in this case occur at a negative value of X2. 
But this is physically impossible, since X2 is a mass. 
By referring to (5), (7) it will be seen that this case, in which Ai is 
negative for all values of Xi, X2, and in which an equilibrium of the type 
defined by (12) is physically impossible, corresponds to a species 5i in- 
capable of maintaining itself even in the absence of the tax placed upon, 
it by the species 52 feeding upon it. This is a case of minor interest. 
If, on the contrary (12) can be satisfied by a positive value of Ai, so 
that an equilibrium of the type (12) is physically possible, then, evidently, 
by (28), X is a pure imaginary. The solution (23), (24) then takes the 
form of Fourier's series; the process is an undamped oscillation con- 
tinuing indefinitely. 
In this connection, it is interesting to recall a passage in Spencer's 
"First Principles," chapter 22, paragraph 173: 
"Every species of plant and animal is perpetually undergoing a rhyth- 
mical variation in number — now from abundance of food and absence of 
enemies rising above its average, and then by a consequent scarcity of 
food and abundance of enemies being depressed below its average 
amid these oscillations produced by their conflict, lies that average 
number of the species at which its expansive tendency is in equilibrium 
with surrounding repressive tendencies. Nor can it be questioned that 
this balancing of the preservative and destructive forces which we see 
going on in every race must necessarily go on. Since increase of numbers 
cannot but continue until increase of mortality stops it, and decrease 
of numbers cannot but continue until it is either arrested by fertility or 
extinguishes the race entirely." 
A question now arises. Do the curves representing the solution (23), (24) 
dip below the zero axes of Xi, X2? This would mean that one or the other, 
or both, of the species 5i, 52 would become extinct through the violence- 
of the oscillations. 
To answer this question we consider the relation: 
dX^ _ X2{A2X\ — B2) /^QN 
dXi ~ XM1-B1X2) ^ 
which is obtained from (8) and (10) by division. From the periodicitjr 
of Xi, X2 (and, therefore, Xi, X2) it follows that the curve defined in rectan- 
gular coordinates Xi, X2 by (29) is a closed curve. Furthermore, this- 
curve can never cross the Xi axis, for at all points of this axis the first,. 
