TWO NOTES ON THE ELLIPSE 



47 



a\ b\ are semi -conjugate axes, and 6 the angle included by them. 

 Sin ^ is a minimum when a^ 1? is a maximum. This will be the 

 case when o}==V\ for 



{a'—hy + 2 a' h' =«' + *^' = constant. 

 Now 2 o^ h^ is a maximum when 



That is when a^ = J/. 



a^ + h' = a' + h' = 2a' 



2 a" + 1/ 



O} = TT . 



Then we have 



Sin e 



ah ah 



a'h' 



2ah 



a-" + b" 



11. 



On A A^ as a diameter, construct 

 the semi-circumference A F A^^ and 

 the semi-ellipse A B A\ Construct 

 the angle B C E ^ angle A C D. 

 Draw E H perpendicular to A A\ 

 intersecting ellipse in E\ Draw C D 

 and C D^ perpendicular to the tangents at P and P', respectively. 



Let <^ = eccentric angle of P. 

 0' = eccentric angle of P'. 

 a = angle D C A. 



By Fagnani's theorem we have 



B P-A P' = P D 



