EXAMPLES OF MAXIMA AND MINIMA. 243 



If we differentiate the values of -=- and -^ , and after dif- 



dx ay 



ferentiation use the relations which arise from ~ = and 



dx 

 du 



-j- = 0. we shall find 

 dy 



d z u _ h (2ky a b}_ 2k*r a b 

 dx* xv rv ~' 



d 2 u _ Tc (2hx -a-b} 2h*r -a-b 

 dy* yv rv ' 



dxdy v ' 

 Hence the sign of A G W is the same as the sign of 



--- _ 

 r* 



and is therefore the same as the sign of 



Now it may be shewn that if a + b be not zero and a be 

 not equal to b, the sign of the last expression is positive 

 for both the values which r can have. For suppose a + b 



positive ; then we have to shew that -p ^-5- r is positive, 



2i \fl T K j 



that is, we have to shew that r-rra j^r is greater than the 



& \tl "f* K J 



positive root of the quadratic in r. Substitute the positive 

 quantity 2 pr for r in the expression which forms the 

 left-hand member of the quadratic ; we shall obtain a positive 

 result if a and b are unequal ; this shews that p ^ greater 



2i (/I ~T K J 



than the positive root of the quadratic (Algebra, Art. 339). 

 Similarly we may establish the result if a + b is negative. 



Hence the necessary conditions for a maximum or mini- 

 mum are fulfilled. 



R2 



