252 MAXIMA AND MINIMA VALUES 



may be expressed in terms of the remaining m n. The 

 simplest theoretical method of investigating the maxima and 

 minima values of <f> would be to express by means of the 

 given equations the values of n of the variables in terms 

 of the rest, and to substitute these values in </>; thus < 

 would become a function of m n independent variables, 

 and we might proceed to ascertain its maxima and minima 

 values in the manner already given for functions of one, two, 

 or three independent variables. But this method would be 

 often impracticable on account of the difficulty of solving the 

 given equations, and the following method is therefore 

 adopted. 



Suppose x, y, z... all functions of some new variable t, of 

 which consequently <f> becomes a function. Put for shortness 



From the n given equations (1) we deduce 



dF.^ dF^ d ^ Dz+ = o 



(3). 



By solving the linear equations (3) we can express n of 

 the quantities Dx, Dy, Dz... in terms of the remaining 

 m n. Substitute these values in (2), then only m n of the 

 quantities Dx, Dy, Dz ... remain, and we have a result 

 which may be written. 



where X, Y, Z, ... do not involve any of the quantities 

 Dx, Dy, Dz, ... Since, consistently with the given equa- 

 tions, we may consider the m n quantities Dx } Dy, Dz, ... 



