25] EFFLUX OF LIQUIDS. 27 



the ' coefficient of contraction/ If the orifice be simply a hole in 

 a thin wall, this coefficient is found experimentally to be about *62. 



The paths of the particles at the vena contracta being nearly 

 straight, there is little or no variation of pressure as we pass from 

 the axis to the outer surface of the jet. We may therefore assume 

 the velocity there to be uniform throughout the section, and to have 

 the value given by (2), where z now denotes the depth of the vena 

 contracta below the surface of the liquid in the vessel. The rate 

 of efflux is therefore 



The calculation of the form of the issuing jet presents difficulties which 

 have only been overcome in a few ideal cases of motion in two dimensions. (See 

 Chapter iv.) It may however be shewn that the coefficient of contraction must, 

 in general, lie between and 1. To put the argument in its simplest form, 

 let us first take the case of liquid issuing from a vessel the pressure in which, 

 at a distance from the orifice, exceeds that of the external space by the 

 amount P, gravity being neglected. When the orifice is closed by a plate, the 

 resultant pressure of the fluid on the containing vessel is of course nil. If 

 when the plate is removed, we assume (for the moment) that the pressure on 

 the walls remains sensibly equal to P, there will be an unbalanced pressure 

 PS acting on the vessel in the direction opposite to that of the jet, and 

 tending to make it recoil. The equal and contrary reaction on the fluid 

 produces in unit time the velocity q in the mass pqS' flowing through the 

 ' vena contracta,' whence 



PS= P fS' ....................................... (i). 



The principle of energy gives, as in Art. 23, 



P=W ....................................... (ii), 



so that, comparing, we have S' = $S. The formula (1) shews that the 

 pressure on the walls, especially in the neighbourhood of the orifice, will in 

 reality fall somewhat below the static pressure P, so that the left-hand side 

 of (i) is too small. The ratio S'/S will therefore in general be >^. 



In one particular case, viz. where a short cylindrical tube, projecting 

 inwards, is attached to the orifice, the assumption above made is sufficiently 

 exact, and the consequent value for the coefficient then agrees with 

 experiment. 



The reasoning is easily modified so as to take account of gravity (or other 

 conservative forces). We have only to substitute for P the excess of the static 

 pressure at the level of the orifice over the pressure outside. The difference 

 of level between the orifice and the ' vena contracta ' is here neglected *. 



* The above theory is due to Borda (Mem. de VAcad. des Sciences, 1766), who 

 also made experiments with the special form of mouth-piece referred to, and found 

 8/8' = 1-942. It was re-discovered by Hanlon, Proc. Lond. Math. Soc. t. iii. p. 4, 

 {1869) ; the question is further elucidated in a note appended to this paper by 

 Maxwell. See also Froude and J. Thomson, Proc. Glasgoio Phil. Soc. t. x., (1876). 



