Lord Rayleigh's Notes on Hydrodynamics. 441 



In the case of perpendicular incidence cosa = 0, c=l, so 

 that 



COS0=± J 



Vs + 1 

 giving on integration 



# = 2\/s + l + const. 



It appears that the value of x does not approach a finite limit 

 as 6 increases indefinitely. 



LIY. Notes on Hydrodynamics . 

 By Lord Rayleigh, F.R.S.* 



[Plate V.] 



77i6 Contracted Vein. 



THE contraction of a jet of fluid in escaping from a higher 

 to a lower pressure through a hole in a thin plate has 

 been the subject of much controversy. Of late years it has 

 been placed in a much clearer light by a direct application of 

 the principle of momentum to the circumstances of the problem 

 by Messrs. Hanlon and Maxwell f among others. 



For the sake of simplicity the liquid will be supposed to be 

 unacted upon by gravity, and to be expelled from the vessel 

 by the force of compressed air through a hole of area a in a 

 thin plane plate forming part of the sides of the vessel. After 

 passing the hole the jet contracts, and at a little distance 

 assumes the form of a cylindrical bar of reduced area a' . The 

 ratio a' : a is called the coefficient of contraction. 



The velocity acquired by the fluid in escaping from the 

 pressure p is determined, in the absence of friction, by the 

 principle of energy alone. If the density of the fluid be unity, 

 and the acquired velocity v, 



v 2 =2p (1) 



The product of v, as given by (1), and a is sometimes, though 

 very improperly, called the theoretical discharge ; and it dif- 

 fers from the true discharge for two reasons. In the first 

 place, the velocity of the fluid is not equal to v over the whole 

 of the area of the orifice. At the edge, where the jet is free, 

 the velocity is indeed v ; but in the interior of the jet the 

 pressure is above atmosphere, and therefore the velocity less 

 than v. And, secondly, it is evident that the quantity of fluid 

 passing the orifice depends, not upon the whole velocity with 



* Communicated by the Author. 



t Proceedings of the Mathematical Society, November 11, 1869. 



