653.] The magnetic potential at any point on either side of the current-sheet is given, as in Art. 415, by the expression where is the distance of the given point from the element of surface dS, and 0 is the angle between the direction of r, and that of the normal drawn from the positive side of dS. This expression gives the magnetic potential for all points not included in the thickness of the current-sheet, and we know that for points within a conductor carrying a current there is no such thing as a magnetic potential. The value of 2 is discontinuous at the current-sheet, for if is its value at a point just within the current-sheet, and its value at a point close to the first but just outside the current-sheet, Ω, = Ω + 4 π φ, +4π where is the current-function at that point of the sheet. The value of the component of magnetic force normal to the sheet is continuous, being the same on both sides of the sheet. The component of the magnetic force parallel to the current-lines is also continuous, but the tangential component perpendicular to the current-lines is discontinuous at the sheet. If & is the length of a curve drawn on the sheet, the component of magnetic force d1 in the direction of ds is, for the negative side, and for the ds positive side, Q2 аф = +4π ds ds The component of the magnetic force on the positive side there аф ds fore exceeds that on the negative side by 4π At a given point this quantity will be a maximum when ds is perpendicular to the current-lines. On the Induction of Electric Currents in a Sheet of Infinite Conductivity. 654.] It was shewn in Art. 579 that in any circuit dp where E is the impressed electromotive force, p the electrokinetic momentum of the circuit, R the resistance of the circuit, and i the current round it. If there is no impressed electromotive force and no resistance, then dp dt = 0, or p is constant. 656.] PLANE SHEET. 263 Now p, the electrokinetic momentum of the circuit, was shewn in Art. 588 to be measured by the surface-integral of magnetic induction through the circuit. Hence, in the case of a currentsheet of no resistance, the surface-integral of magnetic induction through any closed curve drawn on the surface must be constant, and this implies that the normal component of magnetic induction remains constant at every point of the current-sheet. 655.] If, therefore, by the motion of magnets or variations of currents in the neighbourhood, the magnetic field is in any way altered, electric currents will be set up in the current-sheet, such that their magnetic effect, combined with that of the magnets or currents in the field, will maintain the normal component of magnetic induction at every point of the sheet unchanged. If at first there is no magnetic action, and no currents in the sheet, then the normal component of magnetic induction will always be zero at every point of the sheet. The sheet may therefore be regarded as impervious to magnetic induction, and the lines of magnetic induction will be deflected by the sheet exactly in the same way as the lines of flow of an electric current in an infinite and uniform conducting mass would be deflected by the introduction of a sheet of the same form made of a substance of infinite resistance. If the sheet forms a closed or an infinite surface, no magnetic actions which may take place on one side of the sheet will produce any magnetic effect on the other side. Theory of a Plane Current-sheet. 656.] We have seen that the external magnetic action of a current-sheet is equivalent to that of a magnetic shell whose strength at any point is numerically equal to 4, the current-function. When the sheet is a plane one, we may express all the quantities required for the determination of electromagnetic effects in terms of a single function, P, which is the potential due to a sheet of imaginary matter spread over the plane with a surface-density 4. The value of P is of course P (1) where r is the distance from the point (x, y, z) for which P is calculated, to the point x, y, 0 in the plane of the sheet, at which the element da' dy is taken. To find the magnetic potential, we may regard the magnetic shell as consisting of two surfaces parallel to the plane of xy, the first, whose equation is z = c, having the surface-density, and с the second, whose equation is z = c, having the surface-density с respectively, where the suffixes indicate that z- is put for z 2 с in the first expression, and z + for z in the second. 2 Expanding these expressions by Taylor's Theorem, adding them, and then making c infinitely small, we obtain for the magnetic potential due to the sheet at any point external to it, (2) 657.] The quantity P is symmetrical with respect to the plane of the sheet, and is therefore the same when -z is substituted for z. 2, the magnetic potential, changes sign when -z is put for 2. At the positive surface of the sheet Within the sheet, if its magnetic effects arise from the magnetization of its substance, the magnetic potential varies continuously from 27 at the positive surface to -27 at the negative surface. If the sheet contains electric currents, the magnetic force within it does not satisfy the condition of having a potential. The magnetic force within the sheet is, however, perfectly determinate. is the same on both sides of the sheet and throughout its substance. If a and be the components of the magnetic force parallel to 657.] VECTOR-POTENTIAL. 265 x and to y at the positive surface, and a', B' those on the negative surface аф (6) (7) Within the sheet the components vary continuously from a and which connect the components F, G, H of the vector-potential due to the current-sheet with the scalar potential 2, are satisfied if we make dP F=' dy dP G = H = 0. (9) We may also obtain these values by direct integration, thus for F, Since the integration is to be estimated over the infinite plane sheet, and since the first term vanishes at infinity, the expression is reduced to the second term; and by substituting and remembering that depends on x and y', and not on x, y, z, we obtain d F= dy dx dy, If ' is the magnetic potential due to any magnetic or electric system external to the sheet, we may write for the components of the vector-potential due to this system. (10) (11) 658.] Let us now determine the electromotive force at any point of the sheet, supposing the sheet fixed. Let X and Y be the components of the electromotive force parallel to x and to y respectively, then, by Art. 598, we have If the electric resistance of the sheet is uniform and equal to σ, Χ = συ, Y = σv, (14) where u and v are the components of the current, and if p is the at the positive surface of the current-sheet. Hence, equations (12) where the values of the expressions are those corresponding to the positive surface of the sheet. If we differentiate the first of these equations with respect to x, and the second with respect to y, and add the results, we obtain The only value of which satisfies this equation, and is finite and continuous at every point of the plane, and vanishes at an infinite distance, is ¥ = 0. (19) Hence the induction of electric currents in an infinite plane sheet of uniform conductivity is not accompanied with differences of electric potential in different parts of the sheet. Substituting this value of y, and integrating equations (16), (17), we obtain o dP dP dP' 2π dz dt dt =ƒ (z, t). (20) Since the values of the currents in the sheet are found by |