442 Lord Rayleiglfs Notes on Hydrodynamics. 



which the fluid may be moving- at any point, but upon the 

 resolved part of this velocity in a direction perpendicular to 

 the plane of the orifice. Thus it is only in the middle of the 

 jet that the whole velocity is efficient ; near the edge the mo- 

 tion is tangential ; and consequently this part contributes but 

 little to the discharge. It is certain that the discharge will be 

 considerably less than va, or, which is the same thing, that 

 the jet must undergo considerable contraction before the liquid 

 composing it can move in parallel lines with uniform velocity v. 

 Since the actual discharge is cr'v, the quantity of momentum 

 passing away with the jet in unit time is cr'v 2 , and the force 

 generating this momentum is that necessary to hold the vessel 

 at rest. If the whole of the interior surface of the vessel were 

 subject to the pressure p, this force would have no existence. 

 On account of the orifice the equilibrium of internal pressures 

 is disturbed and a force per is uncompensated. But this is not 

 all. Not only is the pressure that would have acted over the 

 area of the orifice wanting, but there is also a relief of pres- 

 sure on the surface surrounding the orifice corresponding to 

 the velocity with which the fluid there moves. The uncom- 

 pensated force tending to produce recoil may therefore be 

 represented by (cr + &cr)p, where h)cr is a small positive quan- 

 tity ; and if the vessel is to remain at rest, a force of this mag- 

 nitude must be applied to it acting in the direction in which 



the iet escapes. Thus , ~ . , 9 /n , 



J l (<T + 6er)p = a'v 2 ; (2) 



and therefore, by (1), j^^ (3) 



expressing that the coefficient of contraction is greater than \. 

 In the absence of a mathematical solution of the problem it 

 is impossible to estimate the magnitude of oV with any preci- 

 sion; but it is something to know from general principle's that 

 there must be a considerable contraction, and yet that the co- 

 efficient of contraction must exceed one half. However, by a 

 slight modification of the problem it is possible to get rid of 

 the uncertainty arising from the unknown magnitude of her. 

 Suppose the hole in a thin plate to be replaced by a thin 

 parallel tube projecting into the interior of the vessel. If the 

 tube be long enough, the sides of the vessel are sufficiently 

 removed from the region of rapid flow to allow of the pressure 

 acting upon them being treated as constant, while the relief of 

 pressure on the sides of the tube does not add any thing to the 

 forces tending to produce momentum in the jet. Under these 

 circumstances, if a be the area of the section of the tube and 

 a' the area of the section of the jet after contraction, a' = ^er, 

 or the coefficient of contraction is one half exactly. The rigo- 

 rous mathematical solution of this problem, so far as relates to 



