On Maxima and Minima of Functions^ i^c, 91 



.'^+ 1 . . 



'- — - =4. This equation may be easily approximated: but 



ft 



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

 formed into the following: (sec- 9~|cos.(p) (cot. ^9— 10) 

 = fn. From either T)f 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 



■ J 



y = a.h.L J to findy in terms of ^r, first put 2r = I, 



and ^=:a,h.I. -— • When z = ly a {=z. tan. 9, be- 

 cause Vince*s Flux. p. 38. subt. =^^=,4i86338;hence 



y will be found =5 773946. Since y varies as z in similar 



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



suspension must be to ihe 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 = — J- , tbe solidity is as Sx^ + a^, the su- 



perficies as x^ + a^,and since n= f,-^ _- ^ =raax. from 



x^ +a* 



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



I 



r 



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 



! 



