On Maxima and Minima of Functions, ^c, 91 



=4. This equation may be easily approximated: but 



if a table of sines and tangents be preferred, it may be trans- 

 formed into the following: (sec. cp — \cos.(p) (coT.itp— 10) 

 =m From either of these equations, cp appears to be 

 =22° 42' 57". Hence the chain, when it includes a max- 

 imum area, must be so placed that the tangents to its two 

 extremities shall make an angle of 45° 25' 54". And since 



y=«.h.l. > to find y in terms of z, first put z = l, 



l+v'l+ft^ 

 a 



and y = a. h. 1. — — When z = l, a {=z. tan. <p, be- 

 cause Vince's Flux. p. 38. subt. =^^=,4186338; hence 



y will be found =,773946. Since y varies as z in similar 

 figures, whatever z may be, the distance of the points of 

 suspension must be to the length of the chain as ,773946 

 to 1. 



Prob. XII. 



To determine the form of a cup, which, with a given 

 thickness and weight of materials, shall have the greatest 

 possible capacity. 



It is easily shewn that this cup must be some portion of a 

 hollow sphere, terminated at the top by a plane. If the 

 thickness be regarded as inconsiderable, with a given super- 

 ficies, we have to make the solid content a maximum. Let 

 y be the ordinate of the generating circular segment, and let 

 the absciss x be made constant, and =a. Then the radius 



of the sphere = — ^ — , the solidity is as 3x^ + a^, the su- 



perficiesasa;2+a2,andsincen=|,-^^ — -^ =raax. from 



which x=a; or the cup must be hemispherical. 



Prob. XIII. 



To determine the same thing, when the thickness of the 

 vessel (supposed in the form of a spherical segment) is in- 

 Considerable at the bottom, and varies in such a manner as 



