DIFFERENTIAL AND INTEGRAL CALCULUS. 85 



should have -^ = 0. -~ = , and therefore y = is a 

 dx J daf 3a 7 , * 



minimum value of ?/ For cases in which ~ takes the 



o dx . 



form - , this method should always be used, putting a -f x 

 for Xj and b + y for ?/, if a, 6, be the values of X, y, 

 for which ~ = - ) . The corresponding curve is repre- 

 sented in fig. (15). 



In finding the maximum or minimum values of any 

 quantity, care should be taken so to choose the variable in 

 terms of which the quantity is expressed (that is, of which 

 it is a function) j that it may be capable of all values. For 

 instance, if u = F(z}^ and z be itself limited in value in 

 any way, being a function of #, an independent variable 



capable of all values, then u = F(z), and -7- =F' (z), but 



du ,, , . dz , du 



= F'( Z ^ and although -j- = when F (z)=0, this 

 dx dx dz 



may not furnish the true maximum or minimum values, 



which may occur when -^ = 0, for which -7- = ; for in- 

 dx ' dx 



stance^ suppose P (fig. 16) is any point on a fixed circle 

 whose centre is (7, any other fixed point, and we seek 

 the maximum and minimum values of OP. Take CN=x^ 

 CP=a, CO = c, then 



OP' 2 = ON* -f NP* =(c- xY + a* - x* 



= c 2 -f a 2cx = Uj 

 then = 2c which can never change sign. But in this 



case x is not capable of all values being limited to values 

 between a and + a. If we make our independent vari- 

 able the angle OCP(=9), which may have any magnitude 



whatever, we have u = a? + c 2 2ca cos#, and-^ = 2ca sin 0, 



CltV 



which changes from - to + as 6 increases through 0, and 

 from + to as 6 increases through TT, hence 6 = gives 



