Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/166

 This result is the same if we substitute for the forces acting on the electrified surfaces an imaginary force whose potential is $$-\tfrac{1}{2}p$$, acting on the whole volume of the element and soliciting it to move so as to increase $$\tfrac{1}{2}p$$.

If we now return to the case of a figure of finite size, bounded by the equipotential surfaces $$S_1$$ and $$S_2$$ and by the surface of no induction $$S_0$$, we may divide the whole space into elements by a series of equipotential surfaces and two series of surfaces of no induction. The charges of electricity on those faces of the elements which are in contact will be equal and opposite, so that the total effect will be that due to the electrical forces acting on the charges on the surfaces $$S_1$$ and $$S_2$$, and by what we have proved this will be the same as the action on the whole volume of the figure due to a system of forces whose potential is $$-\frac{1}{2}p$$.

But we have already shewn that these electrical forces are equivalent to a tension $$p$$ applied at all points of the surfaces $$S_1$$ and $$S_2$$. Hence the effect of this tension is to pull the figure in the direction in which $$p$$ increases. The figure therefore cannot be in equilibrium unless some other forces act on it.

Now we know that if a hydrostatic pressure $$p$$ is applied at every point of the surface of any closed figure, the effect is equal to that of a system of forces acting on the whole volume of the figure and having a potential $$p$$. In this case the figure is pushed in the direction in which $$p$$ diminishes.

We can now arrange matters so that the figure shall be in equilibrium.

At every point of the two equipotential surfaces $$S_1$$ and $$S_2$$, let a tension = $$p$$ be applied, and at every point of the surface of no induction $$S_0$$ let a pressure = $$p$$ be applied. These forces will keep the figure in equilibrium.

For the tension $$p$$ may be considered as a pressure $$p$$ combined with a tension 2$$p$$. We have then a hydrostatic pressure $$p$$ acting at every point of the surface, and a tension \frac{2p}{} acting on $$S_1$$ and $$S_2$$ only.

The effect of the tension $$2p$$ at every point of $$S_1$$ and $$S_2$$ is double of that which we have just calculated, that is, it is equal to that of forces whose potential is $$-p$$ acting on the whole volume of the figure. The effect of the pressure $$p$$ acting on the whole surface is by hydrostatics equal and opposite to that of this system of forces, and will keep the figure in equilibrium.

107.] We have now determined a system of internal forces in