OF A FUNCTION OF TWO INDEPENDENT VARIABLES. 237 



therefore 2 (x 3 - y 3 ) = xy (y - x) ; 



therefore 2 (x y) (x* + xy + y*)=xy(y x): 

 either then x = y, 



or 2ic B + 3icy + 27/ 2 =0. 



The latter leads to an impossible result ; the former gives 



a 



x = y = n> - 

 V 3 



d 2 u 2a 3 



dxdy 



therefore -7-5 -7-5 ( -j y- ) is positive when x and y have 

 dx ay \dx ay) 



the assigned values, and -^ is positive ; hence u is then a 

 minimum. 



2. Let u = cos x cos a + sin x sin a cos (y /3), 



-v- = sin x cos a + cos x sin a cos (y /3), 



^ M / ox 



5- = sm a sin a; sm (y 8). 



dy 



Hence -=- vanishes when y = /3, and then -=^ becomes 

 dy dx 



sin (a x), and vanishes when a; = a. 



Also -7-2 = cos a; cos a since sin a cos (y /S), 



dxdy 



= cos x sn a sn 



u . . n 



-7-j = sin a sin a cos (y p). 



