214 LECTURE XXIII. 



cases the retardation arising from friction is so considerable as to cause a 

 still greater deviation from the quantity which would be discharged by a' 

 shorter pipe in the same time. 



When the aperture through which a fluid is discharged, instead of being 

 every way of inconsiderable magnitude, is continued throughout the height 

 of the vessel, and is every where of equal breadth, the velocity must 

 be materially different at different parts of its height ; but we may find the 

 quantity of the discharge, by supposing the whole velocity equal to two 

 thirds of the velocity at the lowest point. And we may find the quantity 

 discharged by an orifice not continued to the surface, but still of consider- 

 able height, by subtracting from the whole discharge of an orifice so 

 continued, that which would have been produced by such a portion of it, 

 as must be shut up in order to form the orifice actually existing. But in 

 this case, the result will seldom differ materially from that which is found 

 by considering the pressure on the whole orifice, as derived from the height 

 of the fluid above its centre. 



When a cylindrical vessel empties itself by a minute orifice, the velocity 

 of the surface, which is always in the same proportion to the velocity of 

 the fluid in the orifice, is, therefore, uniformly retarded and follows in its 

 descent the same law as a heavy body projected upwards, in its ascent ; 

 consequently the space actually described, in the whole time of descent, is 

 equal to half of that which would have been described, if the initial motion 

 had been uniformly continued ; and in the time that such a vessel occupies 

 in emptying itself, twice the quantity of the fluid would be discharged if it 

 were kept full by a new supply. This may be easily shown, by filling two 

 cylindrical vessels, having equal orifices in their bottoms, and while the one 

 is left to empty itself, pouring into the other the contents of two other equal 

 vessels in succession, so as to keep it constantly full ; for it will be seen 

 that both operations will terminate at the same instant. 



A similar law may be applied to the filling of a lock from a reservoir of 

 constant height ; for in all such cases, twice as long a time is required for 

 the effect, as would be necessary if the initial velocity were continued. The 

 immersion of the orifice in a large reservoir has been found to make no 

 difference in the magnitude of the discharge, so that the pressure may 

 always be estimated by the difference of the levels of the two surfaces. 

 Thus, when a number of reservoirs communicate with each other by ori- 

 fices of any dimensions, the velocity of the fluid flowing through each 

 orifice being inversely as the magnitude of the orifice, and being produced 

 by the difference of the heights of the fluid in the contiguous reservoirs, this 

 difference must be every where as the square of the corresponding velocity. 

 But if the reservoirs were small, and the orifices opposite and near to each 

 other, a much smaller difference in the heights of the surfaces would be 

 sufficient for producing the required velocity. The same circumstances 

 must be considered, in determining the velocity of a fluid forced through a 

 vessel divided by several partitions, with an orifice in each ; if the orifices 

 are small in proportion to their distance from each other, and if they &re 

 turned in different directions, each orifice will require an additional pres- 

 sure, equivalent to the whole velocity produced in it : but if the partitions 



