234- MAXIMA AND MINIMA VALUES 



If M=Q ................................ (3), 



u x-u du , du . , 

 both -J- and -r- vanish. 

 ax ay 



From equations (2) we deduce 

 d*u dM ... dU - dM 



rr ,, dV ^ dM , , . . . . , 



-v-5 = V. , h M . -7- = V. -r when (3) is satisfied. 

 dy* dy dy dy 



* rr dM ,, dV , r dM . , . . ,. c , 

 = V. -r~ + M.-j-=V. -r- when (3) is satisfied, 



dxdy ' dx ' dx ' dx 



* ~, dM T., dU TT dM , , ON . ,. j 

 =U.- r +M.- r =U. -j- when (3) is satisfied. 



But , , = , , always ; hence, when (3) is satisfied, 



Now suppose that from M = 0, we find y in terms of x, 

 say y = ty(x}, and substitute in w; we thus make u a function 

 of x only. On this hypothesis 



du _ /du\ /du\ dy 

 dx \dx) \dy) dx 



, by (2), 



= 0, since M = by hypothesis. 



Hence, this substitution of ty (x} for y has reduced u to 



du 

 a constant, since -j- vanishes without our assigning any parti- 



cular value to x. 



