Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/321

 curvature of lines of force in electrostatic fields, had noticed an apparent tendency of adjacent lines to repel each other, as if each tube of force were inherently disposed to distend laterally; and that in addition to this repellent or diverging force in the transverse direction, he supposed an attractive or contractile force to be exerted at right angles to it, that is to say, in the direction of the lines of force.

Of the existence of these pressures and tensions Maxwell was fully persuaded; and he determined analytical expressions suitable to represent them. The tension along the lines of force must be supposed to maintain the ponderomotive force which acts on the conductor on which the lines of force terminate; and it may therefore be measured by the force which is exerted on unit area of the conductor, i.e., εE2/8πc2 or $1⁄2$DE. The pressure at right angles to the lines of force must then be determined so as to satisfy the condition that the aether is to be in equilibrium.

For this purpose, consider a thin shell of aether included between two equipotential surfaces. The equilibrium of the portion of this shell which is intercepted by a tube of force: requires (as in the theory of the equilibrium of liquid films), that the resultant force per unit area due to the abovementioned normal tensions on its two faces shall have the value T(1/ρ1 + 1/ρ2), where ρ1 and ρ2 denote the principal radii of curvature of the shell at the place, and where T denotes. the lateral stress across unit length of the surface of the shell, I being analogous to the surface-tension of a liquid film.

Now, if t denote the thickness of the shell, the area intercepted on the second face by the tube of force bears to the area intercepted on the first face the ratio (ρ1 + t) (ρ2 + t)/ρ1ρ2; and by the fundamental property of tubes of force, D and E vary inversely as the cross-section of the tube, so the total force on the second face will bear to that on the first face the ratio

or approximately