92 On Maxima and Minima of Functions, fyc. 



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

 fluid is concerned, equally strong throughout. 



The thickness, to verify this condition, must be everywhere 

 as the distance from the top. If the ordinate be do, and the 

 height a, as before, it is shewn without difficulty that the solid 

 content ofthe vessel, or the space contained between the inte- 

 I'ior and exterior surfaces (putting t = thickness at bottom) is 

 equal to *^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 v, the space included between 

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

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

 t be constant, winle the capacity varies so as 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 u continues similar to 

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

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

 in the last problem. 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 x and y 

 alone enter. 



Schol. In two cases, the vessel, of which the outside is 

 spherical, and the thickness every where as the distance 

 from the top, will have its interior surface spherical. When 

 it is a hemisphere, the interior surface will be a hemisphere 

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

 dicularly to the horizon, will be every where the same. 

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

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

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

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

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

 the same. 



Prob. XIV, 



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

 its chord is a maximum. 



