MATHEMATICAL NOTES*. 



1. THE area of a polygon of a given number of sides, circum- 

 scribing a given oval figure, will be the least possible when each 

 side is bisected in the point of contact. 



This elegant proposition, given in the Senate-House Problems 

 for 1836, may be easily demonstrated as follows : 



Let AB, BC, CD, be consecutive sides of the polygon. 

 Produce AB, DC, to meet in E; then BC must, by the con- 

 dition of the minimum, be in such a position that EBC is a 

 maximum. 



Eefer the oval to EA, ED, for axes, then the equation to 

 the tangent BC is 



ydx x'dy ydx xdy, 



y and x being the co-ordinates of the point of contact P. 

 Put aj' = 0; 



and so 



Also area of EBC = %x Q y ti sin E. 



(ydx xdy)* . . .. . . . . 



Hence, - , , * - is a maximum (the minus sign is im 



material). 



Differentiate, considering x as independent ; then 

 fydx-xdy 



xy fyx-xy \ = 

 dxdy y \ dy ) 



The last factor only gives a solution ; 



.. ydx xdy 

 X = -dy- * = **> 



* Cambridge Mathematical Journal., No. IX. Vol. II. p. 142, May, 1840. 



