208 CASES OF GEOMETRICAL 



Jtas a maximum value = a (1 + e), and a minimum value 

 = a (1 e). 



The reason we do not find these values by the above usual 

 analytical process is this. In the ordinary theory of maxima 

 and minima the function is considered to be expressed in 

 terms of an independent variable which may assume all possi- 

 ble values. But in the example above r is not an independent 

 variable ; its values are limited to those found by ascribing 

 all possible values to 6 in the equation 



r = 



1 + e cos ' 



Since r is thus a function of 6, we may consider p 

 which is a function of r to be also a function of 9. Hence 



-~f. = - 7 - - , and this may be made = if we can make 

 at? ar da 



dr 



-TQ = 0. This we can do, and thus p has a maximum or 



minimum value at the same time as r has. 



Similar remarks apply to other examples. Thus generally, 

 if y = < (x), where x is not susceptible of all possible values, 



it may be impossible to make = 0, and thus there may be, 



apparently, no maximum or minimum value of y. But in this 

 case, if x can be expressed in terms of some variable which 



dx 



can assume all possible values, we must put -^ = 0, which 



at/ 



dv 



makes -f n = 0, and thus we determine simultaneous maxima 

 av 



or minima ralues of x and y. 



Example. To find the maximum and minimum length 

 of the straight line drawn to a circle from a given external 

 point. 



Take the axis of x passing through the centre of the circle 

 and the given external point, the former being the origin. Let 

 o = the radius of the circle, c = the distance of the given 

 point (A say) from the centre ; and let x be the abscissa of a 

 point P on the circumference ; then AP Z = c 2 + a 2 2cx. 



