We may describe, through all the points in an electric field which have the same potential, surfaces called equipotential surfaces, and these will be everywhere perpendicular or orthogonal to the lines of electric force. Let us assume the field divided up into tubes of electric force as already explained, and these cut normally by equipotential surfaces. We can then establish some important properties of these tubes and surfaces. At each point in the field the electric force can have but one resultant value. Hence the equipotential surfaces cannot cut each other. Let us suppose any other surface described in the electric field so as to cut the closely compacted tubes. At each point on this surface the resultant force has a certain value, and a certain direction inclined at an angle [theta] to the normal to the selected surface at that point. Let dS be an element of the surface. Then the quantity E cos [theta]dS is the product of the normal component of the force and an element of the surface, and if this is summed up all over the surface we have the total electric flux or induction through the surface, or the surface integral of the normal force mathematically expressed by [int]E cos [theta]dS, provided that the dielectric constant of the medium is unity. We have then a very important theorem as follows:--If any closed surface be described in an electric field which wholly encloses or wholly excludes electrified bodies, then the total flux through this surface is equal to 4[pi]- times the total quantity of electricity within it.[16] This is commonly called Stokes's theorem. The proof is as follows:--Consider any point-charge E of electricity included in any surface S, S, S (see fig. 3), and describe through it as centre a cone of small solid angle d[omega] cutting out of the enclosing surface in two small areas dS and dS' at distances x and x'. Then the electric force due to the point charge q at distance x is q/x, and the resolved part normal to the element of surface dS is q cos[theta]/x². The normal section of the cone at that point is equal to dS cos[theta], and the solid angle d[omega] is equal to dS cos[theta]/x². Hence the flux through dS is qd[omega]. Accordingly, since the total solid angle round a point is 4[pi], it follows that the total flux through the closed surface due to the single point charge q is 4[pi]q, and what is true for one point charge is true for any collection forming a total charge Q of any form. Hence the total electric flux due to a charge Q through an enclosing surface is 4[pi]Q, and therefore is zero through one enclosing no electricity. [Illustration: FIG. 3.]