86 On Maxima and Minima of Functions, ^c. 



Prob. V. 



The solidity of a cone being given, to determine when 

 the inscribed sphere is a maximum. 



A sphere inscribed in a cone will have the same radius 

 with that of a circle inscribed in its vertical trianscular sec- 

 tion. But the radius of a circle inscribed in an isosceles 

 triangle, if a denote half the base and x the height, 



Since the sphere is a maximum when its 



radius is such, u may be taken equal to this radius, if n is 

 made =|. Since then v^ the solidity, varies as or, we have 



to make ——- z=^-=rx^y or — — ~ max. By 



taking the differential coefficient =0, ~x (a + v^^^ +x^ ) 



5 



jns . 



~ , or by reduction, ava2 + x»=|x« — aS 



^^^^_ _ -r- p 



whence ■x=za.2\f2. 



By substitution, the rad. of the inscribed sphere =j^ 



and the diameter of the sphere appears to be a third pro- 

 portional to the diameter of the' base, and the height of the 



a 



~5 



3 



Cfcne. The content of the sphere is l^f-p^ and that of 



the cone is ^fn^ y/sy hence when the sphere is a icwMW' 

 mum, the cone Is double the sphere. 



The foregoing may suffice as a specimen of the applica- 

 tion of this method to problems respecting lines of the first 

 order. 



Prob. Tf, 



- - r ■ 



The area of the parabola, between the curve and a dou* 

 fcle ordinate, being given, it is required to find the relatiofi 

 of the absciss and ordinate, when the inscribed circle is a 



maximum. 



Let the ftbsciss a: be made constant and ^=a ; put the or- 

 dinate, as usual, =y ; then the radius of the inscribed cir- 

 cle is easily determined, by a figure drawn for the purpose, 



i 



