VALIDITY OF THE PRINCIPLE OF LE CHATELIER. 



27 



8xi 5?/i 



a system of linear equations, which we solve for —7 , -77 and obtain 



oh 0(r 



dxi 



(23) 



and a similar expression for 



^yi 

 dG 



Condition of Stability. A general solution of (18), (19) can be 

 written ^ in the form of exponential series 



x= Po-\- Pic^'' + Poe^'' + Pnc^^'' + 

 where Xi X2 are the roots of 



(24) 



(25) 



A(X) 



\dx 



dh (dh_^) 

 dx \dy J 



= 



(26) 



The condition for stability '^ of the equilibrium is that the real parts 

 of all the roots X are negative. This in turn demands that the abso- 

 lute term A(0) be positive. But this absolute term is, evidently, 



A(0) = 



(27) 



6 A. J. Lotka, Proc. Am. Ac, 1920, p. 139. 



7 Idem, loc. cit., p. 144; Hurwitz, Math. Ann., 1875. vol. 46, p. 521 ; Blondel, 

 Jl. de Physique, 1919, pp. 117, 153. 



