its attachment to the balance P1=9.23 grams. Greatest weight required to balance the attraction across the glass plate P2=9.35 grams. Hence the magnetic attractive force f12=0·12 × 981= 118 dynes.

59. Dimensions of physical quantities; dimensions of force. Surfaces are said to be of two dimensions in length, while volumes are three-dimensional. We may express this by writing the number of dimensions as an exponent over a symbol representing the nature of the fundamental unit; thus if [L] stands for some quantity of the nature of a length: The dimensions of area are [L]2

The dimensions of volume are [L]3

We have corresponding analogous expressions for the dimensions of all derived units, formed by combining the fundamental units in any manner whatever. Thus :

The dimensions of velocity are [L]/[T]=[L] [T]1 where [T] stands for some quantity of the nature of an interval of time; and again

The dimensions of acceleration are [L] [T]'/[T]= [L] [T]2, because it is a velocity (increase of velocity) divided by an interval of time.

The dimensions of force are [M] [L] [T]2 where [M] stands for some quantity of the nature of a mass; a force being measurable as the product of a mass and an acceleration.

60. Decrease of magnetic effect with increase of distance.We may easily perceive that the influence which two magnets exert upon one another becomes less when we interpose between them a greater thickness of a medium whose permeability is small; for example, when the magnets are surrounded by air, and we draw their poles further apart. In experiment 35, if we interpose two or more glass plates between the mutually attracting poles, we can measure in dynes the corresponding changes in the force of attraction. If it were possible to obtain a pole from which

G

the lines of force radiated equally in all directions, the magnetic force due to such a pole at a distant point might be expected to diminish in proportion as the square of the distance increased, for the same number of lines of force are distributed over a greater spherical surface, whose extent is proportional to the square of the radius. Hence the number of lines of force which cut through any given area normal to their direction (for example, the unit of area) varies in inverse proportion to the square of the distance.

It was COULOMB who first established the truth of the law of inverse square by direct measurements upon long thin bar magnets. If the magnetic influence is propagated continuously from place to place through the surrounding medium, it is evident from purely geometrical considerations (fig. 23) that the effect at a

2

FIG. 23

place distant r cm. from the centre of attraction or repulsion will be only 1/2 of the effect at a place whose distance is 1 cm. This law holds good for all agents whose influence is exerted equally on all sides, for example, light, sound, thermal radiation; always supposing that the medium has no absorbing effect on the influence in question.

61. The Newtonian law of universal attraction.-Newton had in 1666 assumed the law of variation as the inverse square of the distance, for the gravitational attraction which all bodies exert upon one another, and which constitutes the weight of bodies at the earth's surface. The truth of the assumption was established when it was found to give correctly the motions of the planets round the sun and of the moon round the earth. The planets are nearly spherical in form, and produce the same gravitational effect

as if their entire mass were concentrated at their centres; we may therefore look upon them as equivalent to attracting points, to which the Newtonian law of attraction is directly applicable. Great attention was attracted to this result, owing to its fundamental importance in dynamical astronomy, and it is not surprising that when bodies were found to exert other forces upon one another, such as magnetic forces, a similar law should have been anticipated. But though the planets may be regarded as attracting points, magnetic bodies must be treated somewhat differently. When a magnet is in the form of a long thin bar, the Newtonian law of the inverse square of the distance holds good for the force exerted by each of its poles. It is not to be concluded, however, that the attraction in this case is due to some special magnetic matter, distinct from ordinary substances.

This conclusion, however, was for a long time accepted, the mathematical theory based upon it reaching a very high development even before Coulomb's time and leading to the discovery of many new relations. The hypothesis is now no longer necessary and a little reflection will show its inadequacy. For the Newtonian law takes account of attractions only, and the forces due to magnetic action involve the consideration of two opposite polarities. We should therefore need to formulate some additional hypothesis, such as that of positive and negative magnetic centres exerting actions at a distance which so long prevailed. The forces due to magnetic bodies can only be regarded as arising from point sources of attraction in certain special cases. Such magnetic systems, with poles giving rise to a radial distribution of lines of force, are only of subordinate importance, for example in some technical applications.

62. Strength of pole.-The analogy between the mutual action of magnets and the attraction of gravitation may be carried a step further. According to NEWTON's law, the attraction between two bodies depends on their masses as well as on the distance between them. A similar relation is to be assumed in the case where two magnets act upon one another; so that we speak of quantities of north-seeking

or south-seeking magnetism, or of magnetic masses.' To these modes of expression we owe certain conceptions which are so convenient in the theory of magnetism that we shall retain them in our more modern exposition of the subject. One such conception is that of the strength of a pole; it may be deduced from experiment without recourse to the hypothesis of magnetic masses acting upon one another at a distance. If we compare with one another the poles of different magnetised bars, by examining their action on one and the same bar, differences will be observed under quite similar conditions, and these can only be due to differences of strength amongst the poles themselves. We may express this result numerically by associating with each pole a corresponding number.

Experiment 36.-The magnetised needle 1 of experiment 35 is suspended from the balance as before, but in place of the needle 2, equally strongly magnetised, take some other needle 3, of the same size, but magnetised to a greater or a less extent, the force of attraction between 1 and 3 being fa1. Let the ratio f21: f31 be denoted by n. Now replace the suspended needle by another needle 4, magnetised to any degree, determine its weight, and find in dynes the attractive forces exerted upon it by the corresponding poles of the needles 2 and 3; call these f24 and f34. If we now calculate the ratio f24f34 we obtain the same value as before; that is, ƒ24: f34=n=f 21: ƒ31.

Numerical example.-f21=118 dynes, as was found in experiment 35; f31=283 dynes; and 283: 118=2.4. The needle 4 whose weight P, is 11-52 grams being now suspended in place of 1, it was found that f=187 dynes and f34=449 dynes; the ratio f24f3=449: 187-24, as before.

4

The numerical value of the attraction involves two coefficients, each of which is a characteristic of one pole alone, and independent of the other pole. In other words, the mutual attraction (or repulsion) of two poles is proportional to the product of two numbers, each of which measures quantitatively the effect of one pole, and is called the strength of that pole.

If the strength of the poles used in the last experiment are

denoted by m1, M2, M3, M4, f21 is proportional to me.m, or, if we please, equal to km,m, where k is a constant quantity. In the same way f31km.m1, so that n=m2: m3. Again if f24=k'm2mş and f34=k'm3m, we have f24f34 m2 :m3=n, as was found by actual measurement.

=

63. Unit strength of pole. In order to be able to deduce the strength of a pole as directly as possible from magnetic forces which it exerts, the unit strength of pole must be expressed in terms of an absolute unit of force, the dyne (§ 57). It will be convenient, therefore, to adopt the following definition: a magnetic pole has the unit strength when it repels an equal pole at a distance of one centimetre with a force of one dyne.

Thus two unit poles one centimetre apart exert on one another a force 1/981 as great as the earth's attraction on one gram of matter at its surface.

If a given pole exerts upon a unit pole at a distance of 1 cm. a force of m, dynes, we say that the strength of the pole in question is m1. If it acts upon a pole whose strength is m2, the force will be equal to m, m, dynes. Now let the distance between the poles be changed from 1 cm. to rem. We shall then have for the force exerted by either pole on the other

In the old theory, the factors m, and m, denoted the 'quantities of free magnetic fluid' at the two poles. The greater the effect which the poles produce, the greater is their strength, and the greater is the number by which the strength is measured. When there is no magnetism, the strength of pole is zero.

Equation (2) is the complete symbolical expression of the socalled Law of COULOMB.

64. Signs of the poles; the unit pole.-The strengths of the poles of a uniformly magnetised bar are equal in magnitude but opposite in sense; the one emits exactly as many lines of force as the other absorbs.