of the bounding surface are our lines of force 11, 12, 13.... A narrow region thus bounded by lines of force is called a tube of force.

MAXWELL calls it a sphondyloid (from the Greek opóvdvλos, a vertebra, so that sphondyloid means essentially a narrow space like that which is filled by the spinal chord). The spinal chord was taken to represent the seat of the force, and hence the choice of the term in question. The name 'solenoid,' which is used in the same general sense by MAXWELL and other English writers, we shall only employ with a very special meaning.

FIG. 24

We can conceive of the entire field as divided up into such tubes of force, which originate at north poles and terminate at south poles, as is the case with the lines of force themselves. Where the lines of force spread out from one another, so that the field-intensity is small, the cross-section of the tubes is greatest; where the lines are crowded closely together, so that the intensity is great, the cross-section of the tubes is least. For the bounding surface of each tube, as well as the space within, consists always of the same lines of force.

82. Flux of force. Since in those parts of the field which contain no magnet-poles no lines of force originate or terminate, it follows that along each tube of force the number of lines cutting through the cross-section is constant. The number of lines in question is thus a quantity characteristic of the tube, and it may be easily calculated. If is the field-intensity at some place through which a given tube of force passes, then is also equal to the number of lines of force which at this place cut perpendicularly across the unit of area. If the cross-section of the tube is w cm.2 (taken perpendicularly to the direction of the lines of force) there will be a lines of force cutting through it.

In describing the properties of the lines of force, we

have already used terms which are borrowed from the subject of fluid motion, such nomenclature being especially suitable because the lines of force, as we follow them through the field, exhibit just that kind of continuity which characterises the flow of an incompressible fluid. The same analogy is used in the following definition :

The product of the field-intensity 5 and the cross-section w of the tube of force is called the flux of force through the tube at the place where H and w are measured:

The magnetic force is not here regarded as something which flows along, but its propagation from place to place through the medium is called the flux of force.

Since is measured by the number of lines of force which cut normally through a square centimetre of surface, F=Hw is the number of lines contained in the tube.

83. Law of the conservation of the flux of force.—Let the flux of force through a tube at a given place be F=w. If the cross-section at some other place is w' (greater than w, let us say), and the strength-of-field ', the corresponding flux of force will be F'=' w'.

If there are no sources' or 'sinks' between the two places, that is to say, no magnet-poles, there will be no gain or loss in the number of lines of force as we pass from one place to the other. The same number of lines which previously intersected the area w is now spread over the area w', so that the number of lines per unit of crosssectional area is changed in the ratio : w', the fieldintensity' having become less in proportion as o' has become greater. Hence

'w'w, or F'=F.

The flux of force along a tube does not change between two cross-sections which include no magnet poles between them.

The flux of force is conserved. The value F= Hw obtained at any one place holds good for the whole tube, F being a magnitude characteristic of the tube in question.

In the hydrodynamical analogue F is the quantity of fluid flowing through a given cross-section per unit time. If the fluid is incompressible, the same quantity passes through each crosssection of a tube of flow'; 5 corresponding to the velocity of the fluid.

84. Maxwell's unit solenoids.-Tubes through which the flux of force is unity are called unit tubes. MAXWELL introduced the name solenoid (owλýv, a tube) in this connection, but we shall reserve the name for the unit tubes just defined, calling them unit solenoids,' or simply solenoids. When the field is mapped out into unit tubes, we obtain a very convenient general view of the distribution of field-intensity. The cross-section of the solenoids need not be circular, as in fig. 24; any strip of space which is bounded laterally by lines of force, and through which the flux of force is unity, is equally a solenoid.

Since the flux of force through a solenoid is unity, its cross-section in cm.2 at any place is the reciprocal of the corresponding field-intensity. Where the intensity is less, the tube is wider, while in places of greater intensity the tube becomes contracted. Again, since a positive pole of strength m emits, according to our conventions, 4πm lines of force, its entire flux of force is also measured by 4πm, and consequently 47m solenoids originate at the pole. Thus the solenoids inform us directly of the values of the various sources and sinks in the field.

B.-Flux-of-force diagrams

We will now carry out for some special cases the division of the field into unit solenoids explained in the last paragraph. If the walls of the tubes are of such a form that the cross-sections can be easily calculated, we have at once a convenient means of estimating the field-intensity at each place. But even then a difficulty arises when we attempt to represent a three-dimensional figure by means of a plane diagram. The simplest case is that in which a diagram representing a sectional view is supposed to be rotated about

a suitable axis, and so to develop a representation of the entire structure of the field.

85. Diagram of a unipolar field.-Let a pole whose strength in absolute measure is m= 24/4π = 1.910 unit be surrounded by a homogeneous medium. In accordance with §72, the number of lines of force emitted is 4πm=24, or the total flux of force proceeding from the pole is equal to 24 units. The space surrounding the magnet-pole must therefore be partitioned into 24 (conical) spaces, through each of which the flux of force is unity. The method of effecting this partition is shown in fig. 25, table I (at the end of the book). Through the pole whose strength is 1.910 unit, a straight line is drawn in any direction to serve as axis, and a circle of any radius is described about m as centre. The axial diameter is now to be divided into as many equal parts as there are units in the flux of force from the pole in our case 24. Through each of the 23 points of division, a straight line is drawn perpendicular to the axis, and produced both ways until it cuts the circle, the points of intersection being joined by straight lines to the centre. If the whole figure were to be rotated about the axis, the circle would describe a sphere, while the radii which we have drawn would describe conical surfaces, having their common vertex at m, and dividing space into equal compartments. The zones which these cones cut out from the surface of the sphere are all equal in area, in accordance with well-known geometrical principles.

Since the flux of force 24, which emanates from the pole m, spreads indifferently in all directions, it follows that each unit of surface of the sphere is traversed by the same flux of force, so that the flux through each of our 24 compartments will be exactly unity.

Thus through each of the 24 conical regions into which the surrounding space is divided by the rotation of the figure, the number of lines of force passing is the same.

In the figure, then, the lines of force represented are just those which form the boundaries between bundles of equal numbers of lines.

86. Estimation of field intensity from the diagram.—If the diagram is drawn exactly to the theoretical scale, with the centimetre for the unit of length, or if we know what the scale of the diagram is (in fig. 25 it is appended), we have immediately the means of determining the field-intensities throughout the region, with the aid of a pair of compasses and a little calculation.

If at any place we measure the breadth b between two adjacent lines of force in the diagram, and the distance r of this place from the axis, the annular strip described by b when the figure rotates about its axis will have an area equal to 2πrb.

The line b which measures the breadth between the two lines of force (compare the dotted lines at the places a and b in fig. 25) meets them very nearly at right angles.

Through this area 2πrb, the flux of force is unity; hence,

At the place a near to the pole m (=1.910 unit) we find r= 34, b=0·8 unit of length, by setting off these lengths against the scale shown in the diagram, by means of a pair of compasses. This gives for the approximate value 0.059 cm.-gr. sec.-'. The direct distance from the pole m is 5.7 cm. ; so that the law of inverse square would give =1.910/(5·7)2=0·059, as before. At the place b, more distant from the pole, we have r=16·2, b= 1.9 cm.; so that =0.0052 cm.-gr. sec., while the direct calculation gives $=1.910/(19.3)2=0·0051.

Our method of evaluating field-intensities from the diagram does not appear in the present case to offer any special advantage; but that is because the relations are so simple. We could equally well measure the distance of the place in question from the pole, and divide the known strength of pole by the square of this distance. The value of the method will, however, become more apparent when we come to deal with more complicated cases, where the field is due to a superposition of the effects due to two or more poles, as in the next example.

At each place in the diagram of a single pole, the