MAXIMA AND MINIMA, 209 



The differential coefficient of this expression with respect 

 to x is 2c, which cannot vanish. But if we put x = a cos 0, 

 we have 



AP 3 = c 2 + a 2 - 2ac cos 0, 



d.AP* . 



-TA = 2ac sm 6 ; 

 da 



and = 0, 6=7r, give the minimum and maximum values 

 respectively of AP 3 . 



In this Example the difficulty would not appear if we had 

 so chosen our axes that x should not be a maximum simul- 

 taneously with AP. Calling b the ordinate of A, c the abscissa 

 of A, and a the radius of the circle, we shall have 



= a 2 



c 2 - 2b 



which has its minimum and maximum values, when 



ac 



X = + o~ 



Another solution of the problem is given in Art. 218. 



The following is an analogous case. Find those conjugate 

 diameters in an ellipse of which the sum is a maximum or 

 minimum. Let r and / be any two conjugate diameters, 

 and u = r + r ! , then u is to be a maximum or minimum, 

 while r 8 + r' 2 = a? + 6 2 = c 2 , say ; 



thus u = T + V( cS y2 )> 



du__ r 



~~7 "^ J- ~ . v ..^ 



// A M /' 1* 1 



W/ ^/ I O / J 



fjnt (** /* 



If -5- be put = 0, we get r 2 = - , and therefore r" = ~ . 

 dr 2 2 



This gives us the equal conjugate diameters, the sum of which 

 we know to be a maximum. If we express r, and therefore r, 

 in terms of some variable which can take all possible values, 

 as for example < the inclination of r to the axis major, we 



shall get an additional result. For -j-. = -7- j-r , and there- 



fore, if -fr = 0, we have also - = 0. But -^ = makes r a 

 d<f> d<j> dq> 



maximum or minimum, and thus we obtain the two principal 

 T. D. C. p 



