M E-A 

 ifj ~ ~~ 2 * negative quantity ; .'. the value x -~ gives a max. : also 



#.r. 2. To divide a given line into two parts x and ^/, so that -f -^ 

 ~ min, Herea* ~y. 



j-. 3. To inscribe the greatest rectangle in a given triangle and pa- 

 rabola. 



Let x ~ that part of the perpendicular measured from the vertex, 

 which determines the required rectangle, a perpendicular, then 



EJC. 4. To inscribe the greatest cylinder in a given cone. 

 Using the same notation $ ~, 



Ex. 5. To inscribe the greatest rectangle in a given ellipse. 



Let x the part of the | ax. maj. measured from the centre, which 



determines the rectangle ; then x - . 

 VV 

 Ex. 6. To find y a max, in the equation (x 3 -f. 3/8)2 a \ xtt 



Here x - , and y = 3 v - . 

 V^ 3V 3" 



When a quantity is a max. or min. it frequently shortens the oper- 

 ation to assume its logarithm a max. or min. Thus to find when* 

 V.rs a x -f b X 3 V in x* is a max. or min. assume log. Vj;* o~r-f 6 

 X s ^/ nt or 3 a max. or min. or log. */x* axJfb -J- log. S V "***" 

 max. or mix. ; hence | X g ^~"<** _ | 3*'^ = ,, 



*_ 



6. If a, b t c, d, &c. be the real roots of the equation -^~ =. 0, taken La 



the order of their magnitude, they will render y a minimum and maxi- 

 mum alternately. 



Cor, If there be m roots equal to <*, and n roots equal to b, then there 

 will be one minimum value of y for the root a, and one maximum for 6, 

 if m and n be odd; and neither max, nor mia, values when they ar* 

 even. 



MEASURES. See Weights (tnd Mewntres, 

 176 K3 



