90 On Maxima and Minima of Functions^ S/'c. 



Prob. XI. 



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 may be the greatest pos- 

 sible. 



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

 the ordinate 1/ and the curve z, is =(a + a;)y — az. The 

 equation of the curve is z^ =2flx -f-a;^ ; hence a-{- x = 

 ^/a'^ +z'^, and the area is Va'^ ^z^. y—az {=u.) If the 

 area were supposed constant, a must be made to varyj 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 z, 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 — . ■ ' 



"^ a^'+z-.dy adz ^ „ , «dz 



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



z^ z^ Va'^+z'^ 



z 4- V a'' -i- z^ \ 

 substitution, (putting for 3/ its equal ch.l >j 



^ h.l. — - — =2z. This equation may be 



/«H.2^ " 



thrown into the following form: . h.l. = 2. 



Now -=-^ =tan2:. of the ansle a contained by a line drawc 



z ax '^ C3 T 



touching the upper extremity of the curve and the absciss 

 produced ; and -y/ — + 1 =3ec. of the same angle. Putting 



this secant =5, we have by substitution, .h.l.f -t=^= j 



