MAXIMA AND MINIMA VALUES OF A FUNCTION. 227 

 Since the quantities h and k are independent, we must have 

 ^_0 ^-0 



, - \f) 7 - V. 



ax dy 



Find values of x and y from these equations, say x = a, 



y = b ; let the values of -j-, . -j =- , -y-= . when these values 

 dx* dxdy df 



are assigned to x and y, be denoted by A, B, G, respectively. 

 We have then by Art. 226, 



-n 1 ( l3 cPv , , d 3 v OZ72 d 3 v 7 d s v 



where M. = j-r \h 3 -7-3 + 3&% , . 7 + 3A^ 2 7 , .. + ^ -7-5 



|_3 { rfcc 3 c?^ dy dx dy* dy" 



x being made = a, and y = b, after the differentiations have 

 been performed. 



If A, B, and C do not all vanish, the sign of 



<f> (a + h, b + Jc) - (j> (a, b) 

 will, when h and Jc are taken small enough, depend on 'that of 



If ACB* be negative, it will be possible, by ascribing 

 a suitable value to ^ , to make the last expression vanish and 



change its sign ; and then < (a, b) is neither a maximum nor 

 minimum value of (j> (x, y). Hence generally we must have 

 A G B 2 positive as a condition for the existence of a maxi- 

 mum or minimum. In this case A and G will have the same 

 sign, and Atf + 2Bhk + Ck z will have the same sign as A or 

 C ; and if that sign be positive, <j> (a, b) is a minimum value 

 of < (x, y], if negative, </> (a, b) is a maximum value. 



We say that generally AC B 2 must be positive; because, 

 in fact, there may be a maximum or minimum value when 

 A C B 2 = 0, as we shall now proceed to shew. 



229. To investigate the additional conditions for the ex- 

 istence of a maximum or minimum when ACB Z = 0. 



IfAC-B* = 0, then 



Q2 



