Page:Elementary Text-book of Physics (Anthony, 1897).djvu/152

138 of efflux, through a small orifice, of any fluid of density $$d$$, from a region in which it is under a constant pressure $$p_{a}$$, into a vacuum.

Torricelli's theorem is shown to be approximately true by allowing liquids to run from an orifice in the side of a vessel, and measuring the path of the stream. If the theorem be true, this ought to be a parabola, of which the intersection of the plane of the stream and of the surface of the liquid is the directrix; for each portion of the liquid, after it has passed the orifice, will behave as a solid body, and move in a parabolic path. The equation of this path is found, as in § 52, to be $$x^2 = - \frac{2v^2}{g}y.$$. Now by Torricelli's theorem we may substitute for $$v^2$$ its value $$2gh$$, whence $$x^2 = 4hy$$. In this equation, since the initial movement of the stream is supposed to be horizontal, the perpendicular line through the orifice being the axis of the parabola, and the orifice being the origin, $$h$$ is the distance from the orifice to the directrix. Experiments of this kind have been frequently tried, and the results found to approximate more nearly to the theoretical as various causes of error were removed.

When, however, we attempt to calculate the amount of liquid discharged in a given time, there is found to be a wider discrepancy between the results of calculation and the observed facts. Newton first noticed that the diameter of the jet at a short distance from the orifice is less than that of the orifice. He showed this to be a consequence of the freedom of motion among the particles in the vessel. The particles fiow from all directions towards the orifice, those moving from the sides necessarily issuing in streams inclined towards the axis of the jet. Newton showed that by taking the diameter of the narrow part of the jet, which is called the vena contracta, as the diameter of the orifice, the calculated amount of liquid escaping agreed far more closely with theory.

When the orifice is fitted with a short cylindrical tube, the interference of the different particles of the liquid is in some degree lessened, and the quantity discharged increases nearly to that required by theory.