MAXIMA AND MINIMA VALUES OF A FUNCTION. 249 



and therefore, since h, k, I are independent, 

 d u _ A du _ n du _ 



-j 0, -y- 0, -j V. 



ax dy dz 



Let values of x, y, z be found from these equations, and 



d 2 u d" 2 u 



when these values are substituted in -y- , -5-5 , . . ., let 



dx dy 



d*u _ . d 2 u _ p d*u _ ~ 

 ~M = ' d& ' dz* = ' 



* _ ., d s u _ , d'*u _ , 



dy dz dx dz dx dy 



The sign of 



$ (x + h, y + k, z+l}-(j>(x, y, z) 



can, with the values of x, y, z just found, be made to depend 

 on that of 



(1). 



Hence, that u may have a maximum or minimum value, 

 the expression (1) must retain the same sign, whatever be the 

 signs and values of h, k, I comprised between zero and fixed 

 finite limits. If we put 



h = sl, k = tl, 

 it follows that 



As* + B?+ C+2A't+2tfs + 2C'st ............ (2), 



must be of invariable sign, whatever be the signs and values 

 of s and t. Multiply (2) by A, and rearrange the terms ; then 



(AS + &+ C't) 2 + (AB- C' 2 ) e + 2 (AA - B'G'} t + AC- B 1 * 



.................. (3), 



must retain an invariable sign. 



Hence, (AB- C' 2 ) f + 2 (A A' - B' C') t + AC-B' 3 must 

 be incapable of becoming negative ; therefore 



AB C' 2 must be positive, and ............ (4), 



(AA-B'CJ less than (AB-C'*} (AC-B*) ...... (5);' 



(4) and (5) are the conditions that must be satisfied in order 

 that u may be a maximum or minimum. Conversely, if they 



