26 INTEGKATION OF THE EQUATIONS IN SPECIAL CASES. [CHAP. II. 



Experiment shews however that the converging motion above 

 spoken of ceases at a short distance beyond the orifice, and that 

 the jet then becomes approximately cylindrical. 



The ratio of the area of the section $ of the jet at this point 

 (called the vena contracta ) to the area S of the orifice is called 

 the coefficient of contraction. If the orifice be simply a hole in 

 a thin wall, this coefficient is found to be about 62. If a short 

 cylindrical tube be attached externally, the value of the coefficient 

 is considerably increased; if, on the other hand, there be attached 

 a short tube projecting inwards, the coefficient is about 5. 



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 surface of the jet. We may therefore assume the 

 velocity there to be uniform, and to have the value given by (14), 

 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 



Jlgz. pS . 



32. The calculation of the form of the issuing jet presents 

 great difficulties, and has only been effected in one or two simple 

 cases. (See Arts. 96, 97, below.) It is, however, easy to shew 

 that the coefficient of contraction cannot (in the absence of fric 

 tion) fall below the value J. For the pressure of the fiuid at the 

 walls of the vessel is approximately equal to the statical pressure 

 P + gpz, except near the orifice, where on account of the velocity q 

 becoming sensible, it is, by (13), somewhat less. Assuming it for 

 the moment to be equal to the statical pressure, we see that the 

 total horizontal pressure exerted on the fluid by the vessel is 



PS + ffpffzdS (15), 



where the integration extends over the area S of the orifice. The 

 horizontal pressure exerted by any one element of the vessel s 

 walls is in fact balanced by that due to an opposite element, ex 

 cept in the case of those elements which are opposite to the orifice. 

 The first term of (15) is balanced by the pressure P of the atmo 

 sphere on the portion of fluid external to the vessel ; so that the 

 total horizontal force acting on the fluid isgpffzd8, or gpzS, if 1$ 

 be the depth of the centre of inertia of the orifice. It is this force 

 which produces the momentum with which the fluid leaves the 



