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



25. We conclude this chapter with a few simple applications 

 of the equations. 



Efflux of Liquids. 



Let us take in the first instance the problem of the efflux 

 of a liquid from a small orifice in the walls of a vessel which 

 is kept filled up to a constant level, so that the motion may be 

 regarded as steady. 



The origin being taken in the upper surface, let the axis of z 

 be vertical, and its positive direction downwards, so that l = gz. 

 If we suppose the area of the upper surface large compared with 

 that of the orifice, the velocity at the former may be neglected. 

 Hence, determining the value of C in Art. 23 (4) so that p = P (the 

 atmospheric pressure), when z = 0, we have 



(I)*- 



At the surface of the issuing jet we have p = P, and therefore 



q* = 2gz .............................. (2), 



i.e. the velocity is that due to the depth below the upper surface. 

 This is known as Torricelli's Theorem. 



We cannot however at once apply this result to calculate the 

 rate of efflux of the fluid, for two reasons. In the first place, the 

 issuing fluid must be regarded as made up of a great number of 

 elementary streams converging from all sides towards the orifice. 

 Its motion is not, therefore, throughout the area of the orifice, 

 everywhere perpendicular to this area, but becomes more and 

 more oblique as we pass from the centre to the sides. Again, 

 the converging motion of the elementary streams must make the 

 pressure at the orifice somewhat greater in the interior of the 

 jet than at the surface, where it is equal to the atmospheric 

 pressure. The velocity, therefore, in the interior of the jet will 

 be somewhat less than that given by (2). 



Experiment shews however that the converging motion above 

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

 the case of a circular orifice) 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 8 of the orifice is called 



* This result is due to D. Bernoulli, /. c. ante p. 23. 



