of the Motion of Incompresdhlc Fluids. 129 



k to z'. Also if P = the atmospheric pressure, and OM = y. 



At the orifice^ = P without sensible error, and z = h Let 

 X = h, and for simplicity sake suppose x' = 0, or the origin 

 to be at M. Then, 



\d^) ^'epresents the value of ^ when h is substituted for x. 

 As the velocity of issuing must after a very short time become 

 uniform, ~ will very soon after the commencement of the 

 motion be = 0. Also - is very small when the orifice is 

 small. 



a 2 ff /i 



2h /dz\ 



This equation is sufficient to show that at the orifice the 

 velocity is always less than that acquired by falling through /i, 

 for — yj^) » which, when the orifice is very small, expresses 



the rate of decrement of the section of the stream in passing 

 through it, is always a positive quantity. 



The experiments of Venturi show that the value of — (—^ 



\dx/ 



is less as h is greater, for he found that by increasing the 

 height of the surface of the fluid above the orifice, the distance 

 of the vena contracta from the orifice was also increased. 



Suppose now k to be the area of the section of the stream 

 at the vena contracta b7ic, the depth of which Mn below the 

 surface is //, For a small distance on each side of the vena 

 contracta the function z will be accurately equal to the section 

 of the stream, and consequently will be a minimum at the 



vena contracta by the definition of it. Hence (j^^ = 

 and u^ = 2g /?, 



exactly in conformity with experience. 



The above is, I believe, the most exact solution of the 

 problem that lias hitherto been given, ns the velocity at the 

 vena contracta has never before been a deduction from theory. 

 It may not be amiss to show that our reasoning will lead to 

 the solution in M. Poissoii's Treatise on Hydrostatics, if 

 conducted upon the supposition that he makes. In general 



N.S. Vol. G. No. 3'2. Ju;r. 1829. S 



