92 On Maxima and Minima of Functions^ ^c. 



to render the vessel, so far as the pressure of a contained 

 fluid is concerned, equally strong ihroughout. 



The thickness, to verify this condition, must be everywhere 

 as the distance from the top. If the ordinate be a?, and the 

 height a, as before, it is shewn without difficuhy that the sohd 

 content of the vessel, or the space contained between the Inte- 

 rior and exterior surfaces (putting t = thickness at bottom) is 

 equal to •Jf/'axrad. of the sphere. Now while the capacity 

 of the vessel is supposed to continue similar to itself by 



making- constant, t the thickness of the bottom, must be 



supposed constant, otherwise tt, the space included between 

 the two surfaces, will be a function of an arbitrary variable 

 quantity, which does not enter into w, the capacity. But if 

 t be constant, uiiile the capacity varies so a^ to continue sim- 

 ilar to itself, the thickness at any other point, which contin- 

 ues similarly situated with regard to the whole surface, will 

 continue constant. Therefore while 7i continues similar to 

 itself in all its dimensions, v varies only in two dimensions ; 

 so that n = |5 as before, and the same result is obtained as 

 in the last problenu The same would be true, should we 

 suppose the thickness from the bottom upwards, to vary as 

 any other function of two dimensions, into which oc and y 



alone enter. 



Schol. In two cases, the vessel, of w^hich the outside is 

 spherical, and the thickness every where as the distance 

 from the top, w ill have Its interior surface spherical. When. 

 it is a hemisphere, the interior surface w-ill be a hemisphere 

 of the same radius^ and the thickness, estimated perpen- 

 dicularly to the horizon, w^ill be every where the same. 

 When it is an entire sphere, the inner surface will also be 

 ail entire sphere, of a radius less than the exterior surface 

 by half the thickness of the bottom ; and the sura of the 

 two thicknesses, contained in any one vertical line, and es- 

 timated in the direction of that Hne, will be every where 

 the samp. 



Pkob. XIV- 



Having given the area of a circular sector, to find when 

 its chord is a maximum. 



