372 Hydrodynamics. 



The theoretical discharge, the discharge through an addi- 

 tional tube, and that through a simple perforation in the side, are 

 as the numbers 16, 13, and 10 nearly. 



480. When an upright cylinder or prismatic vessel is suffer- 

 ed gradually to discharge itself, the velocity of the descending 

 surface of the fluid is to the velocity at the orifice, as the area 

 of the latter is to that of the former, and this is a constant ratio ; 

 consequently the velocity of the descending surface varies as the 

 velocity at the orifice, or as \/s ; that is, the velocity of the 

 descending surface varies as the square root of the space to be 

 described by it ; so that this corresponds exactly to the case of 

 a body projected perpendicularly upward ; whence, as the re- 

 270. tarding force is constant in the instance just referred to, it must 

 be constant also in the case before us. Therefore, when a vessel 

 of the above description is suffered to discharge itself, the velocity of 

 the descending surface and that of the discharged fluid will be uni- 

 formly retarded. 



Suppose a body, urged by a constant force, as that of gravity, 

 to describe a space, as 1 rod, for instance, in the first second ; 

 267t the spaces being as the squares of the times, it will describe 4 

 rods in two seconds, 9 rods in 3 seconds, and so on ; and the 

 spaces described in the first second, second second, &c., will 

 evidently be the differences of these, namely, 1 0, 4 1, 

 9 4, 16 9, &,c., that is, the series of odd numbers,!, 3, 5, 7, 

 9, &c. Accordingly, these numbers, taken in the inverse order, 

 represent the spaces described in equal times by a body uniform- 

 ly retarded ; they represent, moreover, as will be seen from what 

 is above proved, the quantities of fluid discharged in equal times 

 from an aperture in the bottom of a prismatic vessel. Hence, 

 if it were proposed to construct a clepsydra, or water-clock, by 

 means of a prismatic or cylindrical vessel, having an aperture 

 Fig.234. m tne bottom, let the height DB of a vessel which would be 

 completely exhausted in a given time, as 1 2 hours, be divided 

 from the top downward into portions represented by the numbers 

 23, 21, 19, &c., down to 1, which will require the height DB to 

 be divided into 144 equal parts, and these portions 23, 21, &c., 

 will be the spaces through which the upper surface will descend 

 in each successive hour of the exhaustion. 



