160 BUTLER ABT. D 



in the condition 



(t" - n (v" - V) - (p" - V') W' - v') + (mi"-mi'; (wx"-mi') 

 . . . + (m„" - Mn') {mj' - m„') > 0, (227) [170] 



which may be written in the form 



^t^■n - ApAv + A/iiAmi . . . + Aju„Aw„ > 0. (228) [171] 



This must hold true of any two infinitesimally differing phases 

 within the hmits of stabiHty. If we give the value zero to one 

 of the differences in every term except one, it is evident that 

 the values of the two differences in the remaining term must 

 have the same sign, except in the case of Ap and Av, which have 

 opposite signs. Thus we have, for example. 



(-) 



/A^\ 



\Ami/t, V, m^, 



/AM2\ 



\Am2Jt, V, 



>0; 



>0, 

 >0, 



Ml. *"3' 



( 



Afin \ 

 Amn/t. V. 



>0; 



Ml. M2. 



•Mn — 1 



(229) [166] 



[167] 



(230) [168] 

 [169] 



and 



(: 



Av\ 



< 0. 



(231) 



Thus, when v, mi, ... rrin have any given constant values, 

 within the limits of stability, t is an increasing Junction of rj; 

 and when t, v, nh, . . .mn have any given constant values, 

 within the limits of stability, fn is an increasing function of mi, 

 etc. In general, "within the limits of stability, either of the two 

 quantities occurring {after the sign A) in any term of [171] is an 

 increasing function of the other, — except p and v, of which the 

 opposite is true, — when we regard as constant one of the quantities 



