Within the space enclosed, let F, F, &, be continuous and finite, except at a surface of discontinuity, F (x, y, z) where these vector components experience an abrupt change of value; let their values on one side of the surface be simply denoted by F, F, and F, and those on the other side by F, F, and F. Let the positive direction of the normal N, to the closed surface be always that which is drawn inwards; we shall for shortness denote its direction cosines by Į, = cos (N,, X), m, = cos (N,, Y), n, = cos (N,, Z) In like manner we write the direction cosines of the normal to F I = cos (N,, X), m, = cos (N,,Y), n, = cos (N,, Z) If now we draw a straight line parallel to the X axis, this in general must cut the surface S' in an even number of points; first assume that there are only two, and that between these the straight line cuts the surface of discontinuity F. Following this line in the positive direction, it will first enter the surface S' at any point x=x,, at which F2 = F1, and &=&x1, I, d S' = dy dz it will then cut the surface F, where & changes abruptly to the value ; further, Ld F = dy dz the straight line finally will emerge from S' at some point x=2, where & and = We are now in a position to calculate our surface integral; for from the well-known equation cos (F, N) = cos (F, X) cos (N, X) + cos (F, Y) cos (N, Y) + cos (F, Z) cos (N, Z) SURFACE-INTEGRALS AND THEIR PROPERTIES it follows that (2) . ff & cos (F, R,) d S′ = SS &z Lds' + SS &, m, d s' 47 in which all the double integrals are to be taken over S'. Let us now consider one of the members on the right, the first, for example. We observe that, in accordance with the above, If we introduce this, together with the similar expressions for the two other members, we obtain finally from (2) we may call, with Maxwell, the convergence of the vector at the point considered. If now we state equation (3) in words we arrive at the following fundamental theorem. I. The surface-integral of a vector over a given closed surface is, apart from discontinuities, equal to the volume-integral of the convergence of the vector over the whole region enclosed. If there is a surface of discontinuity, a term must be added which on closer consideration may be described as follows: it is the surface-integral of the difference of the normal components of the vector on the two sides of the surface of discontinuity integrated over that portion of the latter which is enclosed by S'. Our straight line parallel to the axis of X may moreover cut the surface S' in more than two points, and number of surfaces of discontinuity may occur without the proof being essentially different. We have reproduced the general proof simplified as any much as possible, for the development here given can scarcely be considered to be as well known as the principle itself.1 § 36. Complex Solenoidal Distribution.-A method suitable for exhibiting the distribution of a vector in space consists in supposing vector lines to be introduced-that is, curves at each point of which the tangent line is coincident in direction with the vector. We have already used this method more than once, having introduced lines of intensity (§ 4) and lines of magnetisation (§ 26). Change of direction along such curves will generally be continuous, but at surfaces of discontinuity they may be abruptly bent, or may terminate. We may further consider the field as divided into wider or narrower tubes fitting closely against each other, the surfaces of which have the curves in question for their generating lines. These figures, which we shall call vector-tubes, will then extend over the whole of the region in question, their direction and their sectional area being in general variable from point to point. These variations may in general be quite unrestricted; such a vector-distribution of the most general kind may be called a complex solenoidal one2 (from owλńv, tube). In dealing with the physical vector quantities which occur in nature we meet with modes of distribution in which the variations of the cross-section, and of the direction of the vector-tubes, are restricted by special relations, to the consideration of which we will now turn our attention. § 37. Solenoidal Distribution.-There is in the first place an important kind of distribution in which the surface-integral of the vector over any given closed surface is zero. A glance at equation (3) shows that this is equivalent to two other necessary 1 For further mathematical details it will be sufficient to refer to Maxwell (Treatise, vol. 1, § 21, Theorem III), which states the theorem, together with the proof in the form given, and names, as the discoverer in 1828, the Russian mathematician Ostrogradsky (Mém. de l'Acad. de St.-Pétersbourg, vol. 1, p. 39, 1831). This important theorem is connected, moreover, with the equation of continuity, as we shall presently see, and may be regarded as a special case of Green's more general theorem which was made known in the same year (Green, Essay on the Application of Mathematics to Electricity and Magnetism, Nottingham, 1828). 2 Sir W. Thomson, Reprint of Papers on Electricity and Magnetism, § 509; from which source most of what immediately follows is taken. SOLENOIDAL DISTRIBUTION 49 and sufficient conditions. In the first place, throughout the space enclosed by the surface we must have (4) = 0 ах მყ az that is, the convergence of the vector at each point must vanish. In the second place, at all surfaces of discontinuity Fz 1, F1⁄2 + m, Fy + n, F; = l, F', + m, F', + n, F. The former equation (4) may be called the volume-equation of continuity, since in hydrodynamics, and in the theories of diffusion and of thermal and electrical conduction, it expresses the condition for the continuity of the corresponding components of flow (Chapter VII.) In like manner equation (5) is to be regarded as the boundary-equation of continuity. We shall now suppose our integration to extend over a surface S' of special form, choosing for this purpose the surface of a finite vector-tube, whose ends are closed by surfaces face-integral S is obviously S1 So FIG. 10 everywhere tangential to the vector, and the latter has consequently no component perpendicular to the surface. The whole surfaceintegral is then seen to be the sum of portions furnished by the end-surfaces, which we designate by [s, and s, But now, from the assumption at the beginning of this paragraph, 8=SS12+SSx+SSx=0 but since √√ ̧ = 0, as explained above, As we previously assumed that the direction of the normal drawn inwards was positive, the sign of the term on the right side is to be reversed if now, on the contrary, we consider the direction of the vector as positive at both surfaces S, and S, (fig. 10). In the mode of distribution here considered, the whole region may be divided up into vector-tubes, which have the following property: 2 II. The surface-integral of the vector is the same over every cross-section of the tube. Such a vector-tube is called a simple solenoid, and the corresponding distribution a solenoidal one. Let us now consider an infinitely thin vector-tube, so that the value of the vector does not vary appreciably over the crosssection, and draw the surfaces S, and S, at right angles to the direction of the tube. If then the values of the vector at the two terminal surfaces are F, and F2, we may write the two portions of the surface-integrals as follows: The product (S) may be called the strength of the infinitely thin vector-tube. The contents of this paragraph may then be expressed in the following proposition :— III. When a vector is distributed solenoidally, the field may be divided up into infinitely thin solenoids of constant strength. Equations (4) and (5) constitute the necessary and sufficient conditions for this mode of distribution. In a thin vector-tube of constant strength ( S), the numerical value of the vector is obviously inversely proportional to the normal cross-section. A solenoid, the strength of which is everywhere unity, may be called a unit-tube; its section is everywhere numerically equal to the reciprocal of the vector; hence the greater the value of the latter, the more unit-tubes will be cut by a given surface. The density' with which the unittubes are crowded together in space furnishes a direct measure of the value of the vector, since the number which falls on a normal surface of unit area is numerically equal to the mean value of the vector over that surface. 6 § 38. Complex Lamellar Distribution. As regards the variation of the direction of the vector from point to point, we premise that in general it is not possible to construct a system |