144 LOTKA. 



themselves, or certain other variables which it may l)e convenient to 

 introduce in their stead, and which serve equally to define the state 

 of the system. 

 Let Fi, Fa, ■ • ■ Y^ be such variables given that 



V = 71 — m (22) 



In place of the primitive system (2) we shall then have a system 

 (ZF, 



dt 



dt 



dY, 



dt 



= MYi, Y2,...Yp; Ai,A,,...A„) 

 = ^2(Fi, }'2,. ..Yv; Ai, A2,...Am) 



= -^XYi, Yo,. ..}'.; Ai, Ao,. . .Am) 



(23) 



the Ai, A-2,. ■ -Am, initial values of .Yi, Xo,. . .X^, functioning as para- 

 meters independent of the time t. 



The system (23) yields to a treatment identical with that applied 

 to the system (2) above, and gives under corresponding conditions a 

 general solution of the form (8) containing (n — m) exponential and 

 (n — m) arbitrary multiplicative coefficients, as required by the 

 conditions of the problem. 



Convergence of the Series (8) and Stability ^^ of the System 

 AT the Origin of the Variables .t. 



To simplify the discussion let us write 



Xy = M. + iv, (24) 



If IX j is zero, Xy is a pure imaginary. 



We need not discuss the case in which the equation A (X) = has 

 zero roots, since these can be eliminated by a change of variables by 

 virtue of the equations of constraint. 



There remain to be discussed the following cases: 



11 A detailed discussion of the question of the existence of series solution 

 of equations of the type (1) will be found in Poincare's memoir "Surles 

 Courbes Definies par les Equations Differentielles," Jl. mathcm. 1886, Ser. 4, 

 V. 2, chapter XVII. See also Encycl. des Sciences Math. t. II, v. 3, fasc. 1. 



