ARTICLES 413 



formulate the condition for equilibrium in precise terms and 

 in form of a minimum or maximum condition. 



We note, first of all, that the condition for equilibrium is 

 given by (6) in the form 



Fi = F2 = . . . = F H — o (6) 



i.e., the equilibrium values of the n variables X lf X 2 . . . X u 

 are defined by the vanishing of n given functions of these 

 variables. 



But evidently the same result can be achieved by stating 

 that some arbitrary function $ (x lf x 2 , . . . x n ) has a mini- 

 mum or a maximum when x lf x 2 , . . . x n have their equi- 

 librium values ; for then we have, for these values, 



d$ _d$ d$ , v 



dx x dx 2 dx n 



i.e., as before, the equilibrium values of the variables x are 

 determined by the vanishing of n functions of these variables. 

 We can go a step further and select the function $ so that 

 it will have a maximum at the origin if, and only if, the equi- 

 librium at the origin is stable. This will, for example, be the 

 case if 



* = A*i*i" + /W + • • . + MA 8 (13) 



where ^ lf /x 2) . . . fi n are the real parts of the roots X. of the 

 equation (11). 



It can be seen by inspection 1 of the solution (10) that a 

 necessary and sufficient condition for stability at the origin 

 is that the real parts of all the roots A, shall be negative — i.e., 

 that the quadratic form $ shall be definite and negative, so 

 that $ has a maximum at the origin. 



As an illustration we may consider an example taken 

 from Sir Ronald Ross's Quantitative Studies in Epidemiology 3 

 — namely, the equations representing the history (evolution) of 

 a system comprising the three species : man — anopheles 

 mosquito — malaria parasite. 



Here Sir Ronald Ross develops the system of differential 

 equations 



^ = k'z\ P -s) +qz= Ffa z') ^ 



& - : - r 04) 



dt 



kz{p' - z') + q'z' = Fiz, z') J 



1 For a rigorous discussion of the conditions of Stability, see Poincare, Jour 

 Mathim., ser. 4, vol. ii, chap, xvii ; also Encyc. des Set. Math.,vo\. ii, pt. 3, fasc. 1 

 3 Nature, October 5, 191 1, p. 466 ; February 8, 1912, p. 497. 



