Hence, R' denoting the total resistance in circuit, the total quantity q' of electricity generated is 4π Nn' A' CR' and instead of (9) we get I=4π Nn' C A'RO The ballistic method of investigation was also used by Prof. H. A. Rowland (Phil. Mag. vol. i., 1875) for the determination of the ideal surface distribution of magnetism on magnets. A thin ring of wire was made just large enough to pass round the magnet experimented on, and was placed in circuit with a ballistic galvanometer. It was then, while encircling the magnet and held with its plane at right angles to the axis of the magnet, slided quickly along the magnet through equal short distances, and the deflection of the needle noted for each motion. The deflections thus obtained gave for thin magnets an approximate comparative estimate of the density at different points along the magnet of the surface distribution of ideal magnetic matter by which the action of the magnet could be produced, and the results were reduced to absolute measure by means of an earth-inductor. This method, used along with Sir William Thomson's method of reduction to absolute measure, gives a very ready means of estimating with much exactness the total quantity of imaginary magnetic matter in one pole or one end of a magnet, whether of bar, horse-shoe, or other shape. The ring, which for the present purpose may be larger, and thick enough to contain any convenient number of turns, is placed at the centre or nearly neutral region of the magnet, and then quickly pulled off and away from the magnet, and the galvanometer deflection (8) noted. A measured current is then sent through the helix, and the deflection (0) produced by suddenly opening the circuit of Y the helix also observed. Let n be the number of turns in the ring of wire, and the total quantity in C.G.S. units of imaginary magnetic matter on the portion of the magnet swept over, then the number of lines of force cut through by each turn of wire in the ring is 4 π Ø, and if R be the total resistance in circuit, the total quantity (q) of electricity generated is 4πno/R. We have therefore and for the helix we get from the calculation above This equation is of course also applicable to the reduction to absolute measure of the results of determinations of magnetic distribution made by the ballistic method. The value of deduced for each deflection divided by the area of the corresponding small portion of the magnet is approximately the surface density of the ideal distribution, the distribution on the end faces being of course included in the end deflections. CHAPTER XI. THEORY OF THE DIMENSIONS OF THE UNITS OF PHYSICAL QUANTITIES. WE have, in p. 70 above, explained the term change ratio of a physical quantity: in order to show clearly the relations of the various absolute units of electrical and magnetic measurement to the units on which they are based, we shall here investigate for each of the principal quantities the formula of dimensions from which the numerical value of the change ratio is to be found in any particular case. A physical quantity is expressed numerically in terms of some convenient magnitude of the same kind taken as unit and compared with it. The expression of the quantity consists essentially of two factors, a numeric,* and the unit with which the quantity measured is compared ; and *The term numeric has been introduced by Prof. James Thomson (Thomson's "Arithmetic," Ed. LXXII., p. 4). It denotes a number, or a proper fraction, or an improper fraction, or an incommensurable ratio. We shall find it convenient to employ it here where we wish to lay stress on the fact that we are dealing with what are essentially numerical expressions. Of course what is actually meant by the conveniently brief expressions "a length, L," "a mass, M," "a force, F," and the like, is simply that L, M, F, &c., denote the numerics which express the respective quantities in terms of the units chosen, that is, are, as we shall say below, the numerics for the quantities in terms of those units. Further in such phrases as the product of mass and velocity," or "the product of charge and potential," and so on, the product (or whatever other function is specified) of the numerics is of course what is intended. If all such expressions were made verbally unexceptionable, the resulting prolixity would be intolerable. the numeric is the ratio of the quantity measured to the quantity chosen as unit. Thus when a certain distance is said to be 25 yards, what is meant is that the distance has by some process been compared with the length, under specified conditions, of a certain standard rod (which length is defined as a yard) and the ratio of the former to the latter found to be 25. The unit of measurement is of course itself capable of being expressed numerically in terms of any unit of the same kind, and in the same way therefore its full expression consists of a numeric and the new unit. Hence if N be the numeric for any physical quantity in terms of any unit, N' in terms of another unit, and n the numeric for the first unit in terms of the second, we have In order therefore to find the numerical expression N' of the quantity in terms of the second unit from its numerical expression N in terms of the first we have to multiply by n, the ratio of the first unit to the second. This numeric has been appropriately called the changeratio for the change from the first unit to the second. The change from N to N' cannot be made unless the change-ration, is known. Each unit may have been arbitrarily chosen without reference to any other unit, and n determined by some process of measurement ; or the units may have been derived from certain chosen fundamental units, and the ratio deduced from the relation of one system of fundamental units to the other. In the measurements described in this work the units employed are entirely of the second kind here referred to. The task before us is to determine the manner in which the various derived units involve the fundamental units, that is, we have to determine for each quantity (p. 180 above) the change-ratio n in terms of the fundamental units. The formula which expresses n for a unit of measurement of any quantity we shall call the Formula of Dimensions or the Dimensional Formula of the quantity. To prevent the necessity for the constant repetition of these terms we shall denote the dimensional formula of any quantity, of which the numerical expression in terms of some chosen unit is denoted by any particular symbol (see p. 326, footnote), by the same symbol inclosed in square brackets. Thus we denote the dimensional formula of the quantity by the symbol [2]. Examples of the values of [Q] will be found in dealing with the various units, to which we now proceed. We shall first consider the definitions and relations of the fundamental units in common use and the derivation from them of the units of other physical quantities. In doing so we shall find the dimensional formula in each case and its numerical values for certain changes of units. For brevity we shall sometimes in what follows call the numerical part of the complete expression of any quantity a numerical quantity of that kind, as this will at once serve to indicate that we are dealing with a numeric, and refer to the quantity with which it is connected. Thus we shall frequently use the phrases, a numerical length, a numerical velocity, and the like to denote the numerics for the quantities in terms of the respective units, whatever these may be, chosen for their expression. In this way L, which denotes the numeric for a length in terms |