244 EXAMPLES OF MAXIMA AND MINIMA. 



Since ACS Z is positive A and C have the same sign, 

 and that sign is the same as the sign of A + C, and therefore 

 the same as the sign of 



r 



If a + 5 is positive this expression is positive or negative 

 according as r is positive or negative ; if a + b is negative 

 it is positive or negative according as r is negative or posi- 

 tive. Thus we can discriminate between the maximum and 

 minimum value of u. 



Two particular cases which have been excepted above 

 remain to be noticed, 



I. Suppose a, = b. Here we shall have 



-T- = 2tf 2 (hx + Tcy a) {hv x (hx + ky a)}, 



-T- = 2tf 2 (kx + ky a}{kv y (hx + 7cy a)}. 

 ay 



If we suppose hx + ky a = we arrive at the case dis- 

 cussed in Art. 235, in which there is not strictly a maxi- 



mum or minimum. If we take the other factors in -y- and 



ax 



du , 

 -7- and put 

 dy 



hv x (hx + Jcya)=Q and kvy (hx + Jcy a) =0, 



we shall obtain 



h k 



x = -- , y = -- ; 

 a' y a' 



these values will be found to make u a maximum. 



The quadratic equation for r, when a = b, has for its roots 



a 1 



T - OT T = * 



h* + V a' 



the former value leads to values of x and y which satisfy 

 hx + Icy a = ; the latter leads to the values 



h Tc 



x = , y= -- . 

 a a 



II. Suppose a + J = 0. The original investigation be- 

 comes inapplicable ; it may be shewn that the only values of 



