will deflect the needles mounted on the axis. As every one of the actions between the respective poles is a statical action, and as the mean of the actions on the nearer pole and the further pole of the needle will be sensibly the same as if each was at the needle's center, the trigonometrical tangent of deflection will be the proportion of the statical force exerted by the magnet to the statical force exerted by the Earth. Now the fact of experiment is, that the deflection produced by the external magnet in the assemblage of needles is exactly the same as the deflection produced in a single needle. And therefore the proportion of the statical force exerted by the external magnet to the statical force exerted by the Earth is the same in both cases. But, as we have seen, the statical force exerted by the Earth is proportioned to the number of needles or to the sum of magnetic energies of the needles; and therefore the statical force exerted by the external magnet is proportional to the sum of magnetic energies of the needles. And the algebraical expression for that statical force must contain that sum of magnetic energies of the disturbed needles as factor. The same rule holds good with regard to gravitation. It may at first appear strange that the pull exerted by a magnet upon several needles is greater than the pull exerted upon a single needle, and that in fact a new equal pull is ready to act upon every new equal needle exposed to it. But the fact is so; and it is analogous to the gravitation-attraction exercised by a planet upon several satellites, in which the force upon one satellite is not diminished by the circumstance that the same planet is acting also upon another satellite. (c) Use the apparatus of Figure 8 as a deflecting apparatus, to deflect from its ordinary position a compass needle. Place the axis of Figure 8 in the direction magnetic E. or W. from the center of the compass: and mount successively upon it one needle, two needles, three needles. If the single needles continue in their combined state each to exercise the same action as when it is alone, so that the whole statical pull on the compass-needle is successively represented by 1, 2, 3, then the trigonometrical tangents of the angles of deflection of the compass-needle will be in the successive proportions of 1, 2, 3. And the fact, in experiment, is so. It follows from this that the statical force exerted by the assemblage of needles is proportional to the sum of the statical forces exerted by each single needle: that is, it is proportional to the sum of the magnetic energies of these needles. And therefore, the expression for the statical force exerted must contain the sum of the energies of the disturbing needles as factor. (This might have been inferred from the conclusion of (b), or vice versa, by assuming the equality of statical action and reaction. But in a matter of such fundamental importance, it appears well to establish each proportion by independent experiment.) (d) Combining the results of (b) and (c), it will be seen that the algebraical expression for the statical force exerted between the two magnetic systems must contain as factor the product of the energies of the two systems. The experiments cited in this Article have been carefully verified by the writer of this Treatise. It is necessary now to fix with precision the units of the different elements which we have to employ. For the unit of time, 1 second of mean solar time is universally adopted: for the unit of measure of length, 1 foot is commonly used in England, and 1 millimètre by the nations which adopt the Metrical system: for the measure of mass, reference is made to weight, and the received units are, 1 grain in England, and 1 milligramme in the Metrical system. For the measure of statical force, it is found convenient to depart from the custom usually followed in mechanical investigations (in which the unit of pressure is considered to be the pressure produced by a unit of mass under the action of terrestrial gravity), and to adopt, instead, that pressure which, acting through the time 1 upon the mass 1, would produce in it the velocity 1. 1 (This unit, in English experiments, is about of the ordinary unit 32.1 of pressure.) This selection of unit of pressure amounts to the same as saying that the unit of accelerative force will be that which produces the velocity 1 in the time 1. SECTION III. ALGEBRAICAL INVESTIGATIONS OF THE ACTION OF ONE MAGNET UPON ANOTHER; THE MAGNETS BEING 14. The disturbing magnet presents the center of one side at right angles to the disturbed magnet (a position which we shall hereafter term "broadside-on"), and the disturbed magnet presents one end to the center of the disturbing magnet (a position which we shall call "endon"); the magnetical energies are supposed to be collected in the poles, and the attractive or repulsive force to vary inversely as the mth power of the distance: to find the angular momentum impressed on the disturbed magnet. Attractions will be represented in the diagrams by continuous lines, repulsions by interrupted lines. In Figure 9, suppose that we require the angular Fig. 9. B momentum which A produces on B. The continuous lines denote attraction; the interrupted lines denote repulsion. Let 2a and 26 be the lengths of the two magnets as measured from pole to pole a and ẞ the magnetic energies at the poles (meaning by this that the attraction or repulsion will be expressed by aß × {distance}"); c the distance between the centers of the magnets, which is supposed to be considerably greater than a or b. Then the distance from the blue pole of A to the red pole of B is α α A b2 (c" — 2ob + a2 + b) * = o ( 1 − 25+ a + b) the attractive force is c the resolved part of this, drawing the red pole of B to a similar term is obtained from the repulsion of the red pole of A on the red pole of B: and the whole angular momentum which they impress on B is |