MATHEMATICAL NOTE 



Suppose we multiply (3) by d</)(g)/dgi, (4) by df(q)/dqi and sub- 

 tract (4) from (3) we obtain 



r = 2 



dgi dgr ^qi ciqr 



^r = 0. 



(5) 



Now we can certainly choose the n — 1 quantities ^2, ?3, . . • • ^n 

 in an arbitrary fashion, since on choosing a set we can adjust 

 the value of ^1 to satisfy (4). It follows that in order to satisfy 

 (5) for any values of ^2, ^3, . . ■ ■ ^n the following relations must 

 be true : — 



bf(q) /d4>(q) bf(q) Idcj^iq) 



bqi J bqi 



bq2 / ^^2 



bf(q) jdckiq) 

 bqn I bqn 



(6) 



since they make all the coefficients of ^2, ^3, . • • • ^n in (5) indi- 

 vidually zero. 



Exactly the same argument shows that if the function 

 f{xi,X2, .... a;„) has a minimum value for the set of values (xr = qr) 

 the same conditions (6) hold. It follows therefore that in 

 order to determine the sets of values of the variables for which 

 the function f(x) is maximum or minimum in value, subject to 

 the condition, (f>{x) = 0, we have to solve the n equations 



bXi bxi 



<t>(x) = 0, 

 bf(x) jb4>{x) ^ 



bX2 I bX2 



bfix) \bct>{x) 

 dXn dXn 



(7) 



Any solution of these equations yields a set of values for "max- 

 min" conditions. 



A special case of this result, which is the one actually required 

 for the considerations arising in Gibbs' Equilibrium of Hetero- 

 geneous Substances * concerns the situation in which the condi- 

 tion imposed on the variables is that their sum should be a 

 constant, i.e. 



Xl + X2 



+ Xn — C = 0. 



See Gibbs, I, pp. 65 and 223. 



