( 248 ) 



CHAPTER XVI. 



MAXIMA AND MINIMA VALUES OF A FUNCTION OF SEVERAL 

 VARIABLES. 



236. LET u = <j>(x, y, z) be a function of three independent 

 variables, of which we require the maxima and minima values. 

 By an investigation similar to that in Art. 224, 



<f> (x + h, y + If, z + 1) - <j> (x, y, z} 



j du ,du jdu 

 dx dy dz 



+ ! i ___!__ i 7.7 u7>7 i 7,7. 



r. J_T ~ 5T3T ,-. 73 i W* j.. j^^" /k j ?_ T" ltn/ 



2 dx' 2 dy* 2 dtf dydz dxdz dxdy 

 + K; 



where It is a function involving powers and products of h, k, I 

 of the third degree, which may be expressed for abbrevia- 

 tion by 



l(,d 7 d 7 



i^ V^T" + &T"*'l' 



[3 [ ax dy 



v denoting < (a; + 0h, y + 6k, z + 6l). 



If we make h, k, I small enough, the sign of 



, z + Z) - <f) (x, y, z] 



will in general depend upon that of the terms involving only 

 the first powers of h, k, I', hence, to ensure a maximum or 

 minimum, we must have 



du ,du ,jdu_ 



h -y- +K -y- + I -T- = 0, 



ax dy az 



