r 



I 



On Maxima and Minima of Functions^ ^c. 87 



y 



j to be =:y~— ; or it varies as ^ay—y^. Let this be made 



u; Vy the area, varies as y, and n = |; hence 2ayi—y^ 

 max. which gives a==f3/j and the radius, by substitution, 



fa. 



\ 



Pkob, VIL 



The solidity of a paraboloid being supposed constant, to 

 determine when the inscribed sphere is a maximum. 



r 



The radius of the inscribed sphere is the same with that 

 of the circle inscribed in the generating parabola, u being 

 again made =i2ay—y^^ since n=:|, and the solidity r varies 



^^y^^--^-^ or 2ay3—y^ must be made a maximum. Pat- 

 ting the differential coefficient =0j a ^2^, or the axis of the 

 paraboloid is equal to its double ordinate. The radius is 



I of the axis. 



Pbob. VIIL 



r « 



Having given the curve superficies of a paraboloid, to 

 find when the solid content is a maximum. 



In this case it will be most convenient to consider the 

 Simple variable quantities on which the functions u and 

 « depend as being the absciss and parameter. Making 

 thS parameter constant and =^, the surface v will be as 



«^ +4ax\l~a^'y the solidity u [^I'rrax^) is as x^ and ft=|. 

 I - Hence — or — that is, - — = mas. whence 



4 ' — ' cj^ /n 



5± J 



If 



i 



.ly 



0, making the 



lb 



3 



2 



a, u is as y, and v as 4 a 2/+^*!— y'; 



y 



^j as before==|; hence 4ay-fy'if y-| — yy= max. which 

 gives «'-^Vya + 3y* =0. This equatioa exhibits the 

 same relatioa between the absciss and parameter with the 



