Page:Elementary Text-book of Physics (Anthony, 1897).djvu/150

136 $$pAv + dAv(\tfrac{1}{2}v^2 + V) = p_{1}A_{1}v_{1} + dA_{1}v_{1}(\tfrac{1}{2}v_{1}^2 + V_{1})$$, whence, since $$A_{1}v_{1} = Av$$, we have $$\frac{p}{d} + \tfrac{1}{2}v^2 + V = \frac{p_{1}}{d} + \tfrac{1}{2}v_{1}^2 + V_{1}.$$

By using again the relation $$A_{1}v_{1} = Av$$, this equation becomes To apply equation (53) to the case of a liquid flowing freely into air from an orifice at $$C$$, we observe that the difference of potential $$(V_{1} - V)$$ equals the work done in carrying a gram from $$C$$ to $$B$$ or equals $$g(h - h_{1})$$, where $$h$$ represents the height of the surface above $$C$$, and $$h_{1}$$ that of the surface above $$B$$. Further we have $$p_{1} = p_{a} + dgh$$, where $$p_{a}$$ is the atmospheric pressure. At the orifice $$p$$ equals $$p_{a}$$. We have then $$\tfrac{1}{2}v^2 \left(1 - \frac{A^2}{A_{1}^2} \right) = g (h - h_{1}) + gh_{1} = gh$$, whence $$v^2 = \frac{2gh}{1 - \frac{A^2}{A_{1}^2}}\cdot$$ If, now, A becomes indefinitely small as compared with $$A_{1}$$, in the limit the velocity at $$C$$ becomes that is, the velocity of efflux of a small stream issuing from an orifice in the wall of a vessel is independent of the density of the liquid, and is equal to the velocity which a body would acquire in falling freely through a distance equal to that between the surface of the liquid and the orifice.

This theorem was first given by Torricelli from considerations based on experiment, and is known as Torricelli's theorem. Its demonstration is due to Daniel Bernoulli.

We may apply the general equation to the case of the efflux of a liquid through a siphon. A siphon is a bent tube which is used to convey a liquid by its own weight over a barrier. One end of the siphon is immersed in the liquid, and the discharging end, which must be below the level of the liquid, opens on the other side of