64 Journal of the Mitchell Society [December 



W 2 / / a^ 2a , , \ 



4ah 



-7=^— 211' 

 n/ 2 



^ 



a? 2a , 



+ -7= h — h^ 



2 v/2 



2h (a y/T — h) 



which is a positive quantity. 



Also 



a^ —( 2 -^ + -^ h + h2 \ 

 V 2 ^ x/2 / 



a^ 2a 



a- — I 2 - 



(*") 



,j-(-f **">•) 



n/2 



^4 h - 2h^ 



h^ 



J^_ 2a^h- 

 \ 2 v/2 



— 2h (aV 2~ + h) 



V(^-')' 



2h2 



which is a negative quantity. 



a 



£t ... 



Hence x = + "7^ is a critical value, which gives rise to a maximum. 



V 2 



As before, we find for the value of the maximum area, 2ab. 



Method III 



We may deal with the function as an implicit function in the 

 variables x and y, subject to the condition that the given point lie 

 upon the required ellipse. 



