4 14 Lord Rayleigh's Notes on Hydrodynamics. 



The values of^ corresponding to the boundaries of the jet are 

 and ir ; and the stream-line which passes symmetrically 

 through the middle of the orifice is ^ = |7r, for which value of 

 -v^ J is purely imaginary. For the stream-line \jr = 0, 



$=e-*+s/e-W—l (9) 



When cj> is negative in (9) ? is wholly real and positive, so that 

 this part of the stream-line is parallel to the axis of x, and 

 answers to the bottom of the vessel up to the edge of the ori- 

 fice. When </> is positive J is complex, but its modulus is 

 unity. This part therefore corresponds to the free boundary. 



The width of the jet after contraction is ir, since the velocity 

 is unity ; and the total flow between the stream-lines yjr = and 

 yjr = 7r is measured by the difference of the values of i/r. 



In equation (9) the real part of f (<£ positive) is cos 0, 

 where is the angle between the direction of motion at any 

 point and the axis of x ; so that the intrinsic equation to the 

 boundary is j v 



cos6= d £=e-% ...... (10) 



no constant being added if s be measured from the edge of the 

 orifice where cos = 1. 



From (10), by integration, 



x=l-e~ 8 , (11) 



if the origin of x be taken at the edge of the orifice, where 

 s = 0. This equation determines the width of the aperture. 

 When s = co , x = l, which corresponds to the abscissa of the 

 boundary of the jet after contraction ; and, as we have already 

 seen, the width of the jet itself is it. Accordingly the whole 

 width of the aperture is 2 + 7r, and the coefficient of contrac- 

 tion 7T : 2 + 7T. 



The numerical value of 7r:2+7r is *611, agreeing very 



nearly with the coefficient of contraction found by observation. 



From (10), 



dy /i : 



oTs=* /1 - e 



whence 



if the origin of y be taken at s = 0. 



If we eliminate s between (11) and (12), we get as the 

 equation of the curve in Cartesian coordinates, 



/« 5 11 1 +^ / 2^P ri3 s 



y= v^-^-ilog^ V 2^3' • • • • W 



* Equations (11) and (12) are given by Kirchhoff. 



