loss of energy of visible motions. Inevitable other forms of resistance; among which are included (1) mutual friction between solids sliding upon one another; (2) resistances due to the viscosity of fluids, or imperfect elasticity of solids; (3) resistances due to the induction of electric currents; (4) resistances due to varying magnetization under the influence of imperfect magnetic retentiveness. No motion in nature can take place without meeting resistance due to some, if not to all, of these influences. It is matter of every day experience that friction and imperfect elasticity of solids impede the action of all artificial mechanisms; and that even when bodies are detached, and left to move freely in the air, as falling bodies, or as projectiles, they experience resistance owing to the viscosity of the air. Effect of tidal friction. The greater masses, planets and comets, moving in a less resisting medium, show less indications of resistance*. Indeed it cannot be said that observation upon any one of these bodies, with the exception of Encke's comet, has demonstrated resistance. But the analogies of nature, and the ascertained facts of physical science, forbid us to doubt that every one of them, every star, and every body of any kind moving in any part of space, has its relative motion impeded by the air, gas, vapour, medium, or whatever we choose to call the substance occupying the space immediately round it; just as the motion of a rifle bullet is impeded by the resistance of the air. 276. There are also indirect resistances, owing to friction. impeding the tidal motions, on all bodies (like the earth) partially or wholly covered by liquid, which, as long as these bodies move relatively to neighbouring bodies, must keep drawing off energy from their relative motions. Thus, if we consider, in the first place, the action of the moon alone, on the earth with its oceans, lakes, and rivers, we perceive that it must tend to equalize the periods of the earth's rotation about its axis, and of the revolution of the two bodies about their centre of inertia; because as long as these periods differ, the tidal action on the Newton, Principia. (Remarks on the first law of motion.) "Majora autem Planetarum et Cometarum corpora motus suos et progressivos et circulares, in spatiis minus resistentibus factos, conservant diutius." Q M tidal earth's surface must keep subtracting energy from their motions. Effect of To view the subject more in detail, and, at the same time, to friction. avoid unnecessary complications, let us suppose the moon to be a uniform spherical body. The mutual action and reaction of gravitation between her mass and the earth's, will be equivalent to a single force in some line through her centre; and must be such as to impede the earth's rotation as long as this is performed in a shorter period than the moon's motion round the earth. It must therefore lie in some such direction as the line MQ in the diagram, which represents, necessarily with enormous exaggeration, its deviation, OQ, from the earth's centre. Now the actual force on the moon in the line MQ, may be regarded as consisting of a force in the line MO towards the earth's centre, sensibly equal in amount to the whole force, and a comparatively very small force in the line MT perpendicular to мо. This latter is very nearly tangential to the moon's path, and is in the direction with her motion. Such a force, if suddenly commencing to act, would, in the first place, increase the moon's velocity; but after a certain time she would have moved so much farther from the earth, in virtue of this acceleration, as to have lost, by moving against the earth's attraction, as much velocity as she had gained by the tangential accelerating force. The effect of a continued tangential force, acting with the motion, but so small in amount as to make only a small deviation at any moment from the circular form of the orbit, is to gradually increase the distance from the central body, and to cause as much again as its own amount of work to be done against the attraction of the central mass, by the kinetic energy of motion lost. The circumstances will be readily understood, by considering this motion round the central body in a very gradual spiral path tending outwards. Provided the law of the central force is the inverse square of the distance, the tangential component of the central force against the motion will be twice. as great as the disturbing tangential force in the direction with the motion; and therefore one-half of the amount of work done loss of energy of visible motions. Tidal friction. Inevitable against the former, is done by the latter, and the other half by kinetic energy taken from the motion. The integral effect on the moon's motion, of the particular disturbing cause now under consideration, is most easily found by using the principle of moments of momenta. Thus we see that as much moment of momentum is gained in any time by the motions of the centres of inertia of the moon and earth relatively to their common centre of inertia, as is lost by the earth's rotation about its axis. The sum of the moments of momentum of the centres of inertia of the moon and earth as moving at present, is about 4:45 times the present moment of momentum of the earth's rotation. The average plane of the former is the ecliptic; and therefore the axes of the two momenta are inclined to one another at the average angle of 23° 27', which, as we are neglecting the sun's influence on the plane of the moon's motion, may be taken as the actual inclination of the two axes at present. The resultant, or whole moment of momentum, is therefore 5·38 times that of the earth's present rotation, and its axis is inclined 19° 13′ to the axis of the earth. Hence the ultimate tendency of the tides is, to reduce the earth and moon to a simple uniform rotation with this resultant moment round this resultant axis, as if they were two parts of one rigid body: in which condition the moon's distance would be increased (approximately) in the ratio 1 : 146, being the ratio of the square of the present moment of momentum of the centres of inertia to the square of the whole moment of momentum ; and the period of revolution in the ratio 1 : 1·77, being that of the cubes of the same quantities. The distance would therefore be increased to 347,100 miles, and the period lengthened to 48:36 days. Were there no other body in the universe but the earth and the moon, these two bodies might go on moving thus for ever, in circular orbits round their common centre of inertia, and the earth rotating about its axis in the same period, so as always to turn the same face to the moon, and therefore to have all the liquids at its surface at rest relatively to the solid. But the existence of the sun would prevent any such state of things from being permanent. There would be solar tides-twice high water and twice low water-in the period of the earth's revolution relatively to the sun (that is loss of visible Tidal to say, twice in the solar day, or, which would be the same Inevitable thing, the month). This could not go on without loss of energy energy of by fluid friction. It is easy to trace the whole course of the motions. disturbance in the earth's and moon's motions which this cause friction. would produce*: its first effect must be to bring the moon to fall in to the earth, with compensation for loss of moment of momentum of the two round their centre of inertia in increase of its distance from the sun, and then to reduce the very rapid rotation of the compound body, Earth-and-Moon, after the collision, and farther increase its distance from the Sun till ultimately, (corresponding action on liquid matter on the Sun having its effect also, and it being for our illustration supposed that there are no other planets,) the two bodies shall rotate round their common centre of inertia, like parts of one rigid body. It is remarkable that the whole frictional effect of the lunar and solar tides should be, first to augment the moon's distance from the earth to a maximum, and then to diminish it, till ultimately the moon falls in to the earth and first to diminish, after that to increase, and lastly to diminish the earth's rotational velocity. We hope to return to the subject later, and to consider the general problem of the motion of any number of rigid bodies. or material points acting on one another with mutual forces, under any actual physical law, and therefore, as we shall see, necessarily subject to loss of energy as long as any of their mutual distances vary; that is to say, until all subside into a state of motion in circles round an axis passing through their centre of inertia, like parts of one rigid body. It is probable *The friction of these solar tides on the earth would cause the earth to rotate still slower; and then the moon's influence, tending to keep the earth rotating with always the same face towards herself, would resist this further reduction in the speed of the rotation. Thus (as explained above with reference to the moon) there would be from the sun a force opposing the earth's rotation, and from the moon a force promoting it. Hence according to the preceding explanation applied to the altered circumstances, the line of the earth's attraction on the moon passes now as before, not through the centre of inertia of the earth, but now in a line slightly behind it (instead of before, as formerly). It therefore now resists the moon's motion of revolution. The combined effect of this resistance and of the earth's attraction on the moon is, like that of a resisting medium, to cause the moon to fall in towards the earth in a spiral path with gradually increasing velocity. VOL. 1. 17 loss of Inevitable that the moon, in ancient times liquid or viscous in its outer layer if not throughout, was thus brought to turn always the same face to the earth. energy of visible motions. Tidal friction. 277. We have no data in the present state of science for estimating the relative importance of tidal friction, and of the resistance of the resisting medium through which the earth and moon move; but whatever it may be, there can be but one ultimate result for such a system as that of the sun and planets, if continuing long enough under existing laws, and not disturbed by meeting with other moving masses in space. That of the solar result is the falling together of all into one mass, which, although rotating for a time, must in the end come to rest relatively to the surrounding medium. Ultimate tendency system. Conserva tion of energy. 278. The theory of energy cannot be completed until we are able to examine the physical influences which accompany loss of energy in each of the classes of resistance mentioned above, § 275. We shall then see that in every case in which energy is lost by resistance, heat is generated; and we shall learn from Joule's investigations that the quantity of heat so generated is a perfectly definite equivalent for the energy lost. Also that in no natural action is there ever a development of energy which cannot be accounted for by the disappearance of an equal amount elsewhere by means of some known physical agency. Thus we shall conclude, that if any limited portion of the material universe could be perfectly isolated, so as to be prevented from either giving energy to, or taking energy from, matter external to it, the sum of its potential and kinetic energies would be the same at all times in other words, that every material system subject to no other forces than actions and reactions between its parts, is a dynamically conservative system, as defined above, § 271. But it is only when the inscrutably minute motions among small parts, possibly the ultimate molecules of matter, which constitute light, heat, and magnetism; and the intermolecular forces of chemical affinity; are taken into account, along with the palpable motions and measurable forces of which we become cognizant by direct observation, that we can recognise |