On Maxima and Minima of Functions, S^c. 87 



to be =?/— |- ; or it varies as 2ay — y^. Let this be made 

 =m; V, the area, varies as y, and n = i; hence 2ayh—yl 

 =max. which gives a=|3/, and the radius, by substitution, 



Prob. VII. 



The soHdity of a paraboloid being supposed constant, to 

 determine when the inscribed sphere is a maximum. 



The radius of the inscribed sphere is the same with that 

 of the circle inscribed in the generating parabola, u being 

 again made =2ay-y% since n=^, and the sohdity v varies 

 asy^,-"-^~^' or 2a2/^- 3/^ must be made a maximum. Put- 

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

 paraboloid is equal to its double ordinate. The radius is 

 =f of the axis. 



Prob. VIIL 



Having given the curve superficies of a paraboloid, te 

 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 

 V depend as being the absciss and parameter. _ Making 

 the parameter constant and =a, the surface v will be as 

 a^ -^-Aaxl^-a^', the soUdity u {=:L'ffax^) is as a;^ and n—^. 



Hence -or- that is, — — ^^ = max. whence 



tjf V a,^j^4ax\2 — a^ 



a;2 _i^aa:+3a2 =0, or a? : 



If we had taken the other formula ^^^^=.0, making the 



absciss constant and = a, m is as y, and v as 4 a y+y'l— y ^-; 



y 



w, as before = |; hence 4 ay -f 2/^ II 2/ "3—^3= max. which 

 gives a^ — Yya + 3y2 =0. This equation exhibits the 

 same relation between the absciss and parameter with the 

 last. 



