OF A FUNCTION OF TWO INDEPENDENT VARIABLES. 229 



enough, the sign of (f> (a + h, b + k} <f> (a, V) is the same as 

 the sign of A. But in order that (j> (a, b) may be a maxi*- 

 mum or minimum value the sign of <f> (a + h, b + k) <f> (a, b) 

 must be invariable when h and k are taken small enough. 



Hence we have the condition that the sign of R t when j- is 



K 

 n 



equal to : , and h and k are taken small enough, must be 



j& 



the same as the sign of A. 



If these two additional conditions are satisfied < (a, b} is a 

 maximum value if A be negative, and a minimum value if A 

 be positive. 



230. If .4 = 0, 5 = 0, and (7=0, we must proceed thus: 



< (a + h, b+k)-<j> (a, 6) = .- {Ptf+ ZQtfk + 3Shk z + Tk 3 



LI 



where P, Q, 8, T, stand for the values of -3, 



^-3, , a , , 

 when x = a and y = b, and 



,-, ;4 737 ji 



-ti [-7 \h -T-J + 4/r/fc , + ... + A; 4 -^-^ 

 [ ( aor rfx j dy <i^ 4 J 



a; being made = a, and y = 6, after the differentiations. 



Hence, that < (a, b} may be a maximum or minimum, it 

 is necessary that P, Q, 8, T, should all vanish. Also, R z 

 must be of invariable sign ; but the conditions to ensure this 

 are too complicated to find investigation here. 



231. The following is another method of investigating 

 the conditions that a function of two independent variables 

 may admit of a maximum or minimum value. 



Let u = < (x, y], where x and y are independent : required 

 the maxima and minima values of u. 



If y, instead of being independent of sc, were equal to 

 some function of x, say -^ (x}, then u would be a function 

 of one variable x. We should then have 



du du 



dor 



