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D I N A J P IT R—DIRAN
variation, and so cannot here lead to a definite result without further knowledge of the physical circumstances. And we remark conversely, in passing, that wherever in a problem of physical dynamics we know that the quantity sought can depend on only three other quantities whose dynamical dimensions are known, it must vary as a simple power of each. The additional knowledge required, in order to enable us to proceed in a case like the present, must be of the form of such an equation of simple variation. In the present case it is involved in the new fact that in an actual gas the mean free path is very great compared with the effective molecular radius. On this account the mean free path is inversely as the number of molecules per unit volume ; and therefore the coefficient of viscosity, being proportional to these two quantities jointly, is independent of either, so long as the other quantities defining the system remain unchanged. If the molecules are taken to be spheres which exert mutual action only during collision, we therefore assume (iozmxvyaz, which requires that the equation of dimensions [ML-1!-1]=[MMLT-^tL? must be satisfied. This gives x= 1, i/ = l, s=-2. As the temperature is proportional to mv2, it follows that the viscosity is proportional to the square root of the mass of the molecule and the square root of the absolute temperature, and inversely proportional to the square of the effective molecular radius, being, as already seen, uninfluenced by change of density. If the atoms are taken to be Boscovichian points exerting mutual attractions, the effective diameter a is not definite ; but we can still proceed in cases where the law of mutual attraction is expressed by a simple formula of variation—that is, provided it is of type km2rs where r is the distance between the two molecules. Then, noting that, as this is a force, the dimensions of k must be [M-1LS+1T-2], we can assume x w li,ccm vyk , provided [ML-1!-1] = [Mp[LT-1KM-1Ls+1T_2]“’, which demands and is satisfied by x-w=, y + 2iv = l, y + (s+)w= -1, 2 s+3 s-3 x so that w= y^sZTl’ x=iZi‘ Thus, on this supposition, s—9 2 s yUOCm2s-2£ s-lt/2s-2 where 0 represents absolute temperature. (See Diffusion of Gases.) In electrical science two essentially distinct systems of measurement were arrived at according as the development began with the phenomena of electrostatics or those of electrokinetics. An electric charge appears as an entity having different dimensions in terms of the fundamental dynamical units, in the two cases: the ratio of these dimensions proves to be the dimensions of a velocity. It was found, first by W. Weber, by measuring the same charge by its static and its kinetic effects, that the ratio of the two units is a velocity sensibly identical with the velocity of light, so far as regards experiments conducted in space devoid of dense matter. The emergence of a definite absolute velocity such as this, out of a comparison of two different ways of approaching the same quantity, entitles us to assert that the two ways can be consolidated into a single dynamical theory only by some development in which this velocity comes to play an actual part. Thus the hypothesis of the mere existence of some complete dynamical theory was enough to show, in the stage which electrical science had reached under Gauss and Weber, that there is a definite physical velocity involved in and underlying electric phenomena, which it would have been hardly possible to imagine as other than a velocity of propagation of electrical effects of some kind. The time was thus ripe for the reconstruction of electric theory by Faraday and Maxwell. The power of the method of dimensions in thus revealing general relations has its source in the hypothesis that, however complicated in appearance, the phenomena are really restricted within the narrow range of dependence on the three fundamental entities. The proposition is also therein involved, that if a changing physical system be
compared with another system in which the scale is altered in different ratios as regards corresponding lengths, masses, and times, then if all quantities affecting the second system are altered from the corresponding quantities affecting the first in the ratios determined by their physical dimensions, the stage of progress of the second system will always correspond to that of the first; under this form the application of the principle, to determine the correlations of the dynamics of similar systems, originated with Newton (Principia, lib. ii. prop. 32). For example, in comparing the behaviour of an animal with that of another animal of the same build but on a smaller scale, we may take the mass per unit volume and the muscular force per unit sectional area to be the same for both; thus [L], [M], . . . being now ratios of corresponding quantities, we have [ML"3] = 1 and [ML-1T-2]=1, giving [L] = [T]; thus the larger animal effects movements of his limbs more slowly in simple proportion to his linear dimensions, while the velocity of movement is the same for both. But this is only on the hypothesis that the extraneous force of gravity does not intervene, for that force does not vary in the same manner as the muscular forces. The result has thus application only to a case like that of fishes in which gravity is equilibrated by the buoyancy of the water. The limbs of a land animal have mainly to support his weight, which varies as the cube of his linear dimensions, while the sectional areas of his muscles and bones vary only as the square thereof. Thus the diameters of his limbs should increase in a greater ratio than that of his body,— theoretically in the latter ratio raised to the power ■§, if other things were the same. An application of this principle, which has become indispensable in modern naval architecture, permits the prediction of the behaviour of a large ship from that of a small-scale model. The principle is also of very wide utility in unravelling the fundamental relations in definite physical problems of such complexity that complete treatment is beyond the present powers of mathematical analysis; it has been applied, for example, to the motions of systems involving viscous fluids, in elucidation of wind and waves, by Helmholtz (Akad. Berlin, 1873 and 1889), and in the electrodynamics of material atomic systems in motion by Lorentz and by Larmor. (j. l*.) Dina]pur, a town (with a population in 1891 of 12,204) and district of British India, in the Bajshahi division of Northern Bengal. The earthquake of 12th June 1897 caused serious damage to most of the public buildings of the town. There is a railway station; a Government high school, with 284 pupils in 1896-97 ; and five printing-presses, with one vernacular periodical. The district comprises an area of 4118 square miles. Its population in 1881 was 1,514,346, and in 1891 was 1,555,835, giving an average density of 378 persons per square mile, being the lowest in the plains of the province. Classified according to religion, Hindus numbered 740,442 ; Mahommedans, 802,597 ; aborigines, 10,694; Christians, 511, of whom 30 were Europeans; “others,” 1291. In 1901 the population was 1,569,133, showing an increase of 6 per cent. The land revenue and rates were Bs.l6,26,711; the number of police was 423; the number of boys at school in 1896-97 was 22,489, being 18‘4 per cent, of the male population of school-going age; the registered death-rate in 1897 was 34,72 per thousand. The district is partly traversed by the main line of the Eastern Bengal Railway and by two branch lines. Dinan, chief town of arrondissement, department of C6tes-du-Nord, France, 35 miles east by south of St Brieuc, on railway from that town to St Malo. In a suburb of the town are many English residents. The new lycee (1892)
