VOL. LXXXV.] PHILOSOPHICAL TRANSACTIONS. 469 



gives the velocity of the effluent fluid to be equal to that which a body would 

 acquire by falling in vacuo through a space equal to the depth of the fluid. 

 But the velocity here determined is not that at the orifice, but at a small distance 

 from the orifice ; for the fluid flowing to the orifice contracts the stream, and 

 the velocity being inversely as the area of the section, the velocity continues to 

 increase as long as the stream, by the expelling force of the fluid, keeps dimi- 

 nishing, and when the stream ceases to be contracted by that force, at that sec- 

 tion of the stream called the vena contracta, the velocity is that which a body 

 would acquire in falling through a space equal to the depth of the fluid. If 

 therefore abcc^ep pi. 5, fig. 11, be the vessel, cd the orifice, cmnd the form of 

 the stream till it comes to the vena contracta, then this investigation supposes 

 KBcmnd'EF to be the form of the vessel, and mn the orifice, the fluid flowing 

 through cmnd just as if the vessel were so continued. But as the proposition is 

 to find the velocity of the fluid going out of the vessel, it may perhaps appeal 

 an arbitrary assumption to substitute the orifice mn instead of cd, when no such 

 a quantity as mn appears in the investigation. If however, we grant that the ex- 

 pelling force must act without any diminution till the fluid comes to mn, it 

 seems that from the principles here assumed we ought to substitute mn instead 

 of cd, as otherwise we get the velocity generated by the action of only a part of 

 the force. The conclusion here deduced agrees very well with experiment ; but 

 an application of the same principles to another case differs so widely from 

 matter of fact, as to render it very doubtful how far the principles here applied 

 can be admitted. And if we were to grant the application of the principles here 

 assumed, so far as regards the determination of the velocity, yet the time of 

 emptying a vessel can by no means be deduced from it. 



In order to determine the time of emptying a vessel, we must know both the 

 area of the orifice cd, and the velocity at that orifice. Now the theory gives 

 only the velocity at mn ; and as it gives not the ratio of mn to cd, the velocity at 

 the orifice cannot thence be deduced, and therefore we cannot find the time of 

 emptying. No theory whatever has attempted to investigate the ratio of mn to 

 cd ; it is well known that it is only to be determined by an actual mensuration. 

 When the orifice is very small, Sir Isaac Newton found the ratio to be that of 

 1 to s/2; when the orifice is larger, the ratio approaches nearer to that of 

 equality. We cannot therefore, even in the most simple case, determine, by 

 theory alone, the time in which a vessel will empty itself. 



If ABCD (fig. 12) be a vessel filled with fluid, and a pipe 7«/iri be inserted at the 

 bottom, mn being very small in respect to bc ; then, according to the theory of D. 

 Bernouilli, the fluid ought to flow out of the pipe at rs with the same velocity it 

 would out of a vessel almd through the orifice rs. Now in this latter case, thevelo- 

 city, according to his own principles, varies as the square root of la, and there- 



