90 On Maxima and Minima of Functions^ &fc. 



Pbob. XL 



It is required to suspend a flexible chain, of given length, 

 in such a manner that the area, included by it, and the 

 straight line joining its extremities maybe the greatest pos- 

 sible. 



The area of this curve, contained between the absciss x, 

 the ordinate 3? and the curve z, is ^={a-\-x)y — az. The 

 equation of the curve is z^ =^2ax -^-x^ 'y hence a-\ x 



Va-H-z^^and the area is "J a^ -\-z^ . y — az {=^u.) If the 



lust be made to vary; but 



by making -^ a maximum, or-=^ -5 =:max» a may 



be supposed constant. We might proceed to exterminate 

 y, and to find the differential with one variable r, as in the 

 preceding cases; but in the present case, it will be most 

 convenient to retain y in finding the differential, and to ex- 

 terminate it afterwards. The differential equation, consid- 



siderine: y and z as both variable, is — — — - "t 



^^F+z^Au aAz ^ ^ ^ «d5r 



— -^-f-— —-=0. But dy— — , - -; hence by 



z^ >/a^-^z^ 



substitution, (puttmg tor y its equal en. I »: j 



- ,-- _^-^ h,L— I :=2r. J-his equation may be 



/^2 1^3 « 



2a3 /a^ 



thrown into the following form: - , >hJ, _2. 



Now -=1^ —tans- of the angle a contained bv a line drawc 



fi cic 



touching the upper extremity of the curve and the absciss 



— -f 1 =sec. of the same angle- Putting 



2s3 — 1 . , / 5+! 



this secant =5, we have by substitution, .h.Ll 



\/s2 — 1 



2. Buth.I.f-f^)=ih.l.f!±I)j hence ?il=i.h.L 



