OF A FUNCTION OF SEVERAL VARIABLES. 251 



237. Example. Let u = 



du _ yz (ay x*} u (ay x 2 ) 



dx ~ (a+xf(x~+y} z (y + z) (z + 1} ~ x (a + x) (x + y) ' 



Q . ., , du uxz-*) 



similarly, -7- = 



du u (by z") 



dz z (y + z) (z + b) ' 



Hence, if ay a? = 0, xz y i = 0, and byz 2 = Q,u may be 

 a maximum or minimum : these equations give 



x _y _z _b 

 a x y z* 



*//x v z b\ */b 



therefore each of these fractions = A /(-.-.--) or A /- . 



/ y \a x y z) ^ a 



Call this r ; then 



x=ar, y = xr = ar 3 , z =yr = ar 3 . 



Proceeding to the second differential coefficients of u, we 

 have 



-r-5 -- / - r^ - N -> 

 ax x (a + x) (x+ y) 



the terms included in the &c. being such as vanish when the 

 specific values are assigned to x, y, z. 



2u 2 



Hence A = 



o 



Similarly B, C, ... can be found, and we shall finally arrive 

 at the result that u is a maximum. 



238. Suppose it required to determine the maxima and 

 minima values of a function ^> (x, y, z, ...) of m variables, 

 these variables being connected by n equations, of which the 

 general form is 



F r (x,y,z,...}^0 (1). 



The m variables involved in <f> are of course not all inde- 

 pendent, since by means of the given equations n of them 



