THE LONDON, EDINBURGH, AND DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF [FOURTH SERIES.] JULY 1871. I. On the Universal Powers of Nature and their Mutual Dependence. By L. A. COLDING*. MY Y previous works upon this subject have been so favourably accepted by the Royal Scientific Society, that I am encouraged once more to lay before the Society some inquiries founded upon the principle of the Lost Forces, which I formerly stated; and I am the more happy at having been requested to continue my researches, which have afforded me many pleasant recreations from my other occupations, because the sequel will contain a basis for a series of inquiries which, I am sure, are in many respects not without interest. On former occasions, as is well known, I have in part referred to the intimate connexion which is proved to exist between the powers of nature; in part I have tried to explain the common law according to which the respective powers of nature may be developed from each other; and the correctness of the fundamental principles proposed here has been confirmed by experiments, which I have performed upon the heat developed by friction of solid bodies. I cannot omit the remark that just as it is the various forces of nature connected with the parts of matter which continually have caused and continually will cause the incessant development of the endless variety of different bodies which nature presents at all times, and just as the peculiar character of the bodies is owing to these forces, so the incessant change which, in fact, may be considered to be the characteristic of matter is caused by their mutual effect. But a general view of the * Communicated by Professor Tait. Phil. Mag. S. 4. Vol. 42. No. 277. July 1871. B various energies must evidently call forth the idea that they also are produced and developed for the purpose of disappearing after having performed one or another effect on the particles of matter; for, in the first place, it is well known that every kind of energy (as, for example, energy of heat, mechanical energy, electric energy, &c.) is able to produce all these energies; and, secondly, we know that when quantities of mechanical work, quantities of heat, &c. are produced through certain quantities of work, quantities of heat, &c., then these energies disappear by degrees as new ones are produced. It is a well-known fact, too, that the production of heat through heat, or of quantities of mechanical work through quantities of mechanical work, &c., is in reality nothing but the energy imparted from one system of material particles to another, and no new production, and also that the receiving body can at most merely receive an increment of energy of the same quantity as that which the imparting body loses; on the contrary, keeping to the indistinct view of the energies having acted their parts when certain material results have been produced, we have not as yet formed a clear idea of the general proportion between the acting and the producing forces. Thus, for example, when the mechanical energy contained in a quantity of water falling upon a water-wheel drives a saw-mill, then it produces every moment a certain material result, but the corresponding mechanical energy itself is lost. Or when heat, developed by burning coal under a steam-boiler, moves a corn-mill by means of a steam-engine, then the heat likewise every moment produces a material result, at which the common idea stops; but the energy of heat which has produced this result exists no more, and we say it has become latent. In the same manner, when the electrical current developed by chemical forces is performing a certain work by means of an electromagnetic machine, then its energy disappears during the work, &c. That new forces, as heat, electricity, &c., are developed along with the material work is indeed well known; but this is generally considered a secondary thing. This view has, however, always appeared to me a very unpleasant one; and I think, on the contrary, that the only natural view of this subject is, as I explained before: That the forces by no means vanish in matter, and consequently it must be a general law of nature that the forces, without exception, undergo a mere change when they seem to vanish, and afterwards reappear as active sources of power of the same quantity but under different forms*. If the alleged proposition is correct, it is evident that the * See Die organische Bewegung in ihrem Zusammenhange mit dem Stoffwechsel, by Dr. J. R. Mayer (Heilbronn, 1845); and Helmholtz, Ueber die Erhaltung der Kraft (Berlin, 1847). different kinds of energy, as energy of heat, mechanical energy, the energy produced by the mutual effect of chemical forces, &c., cannot be of different essence, but that all the various kinds must be looked upon as one and the same energy, as, for example, mechanical energy. As heat consists in the motion of the particles of bodies, it follows from this: 1. That the material particles of which bodies consist are in continual motion, even when the particles of the body seem to be at perfect rest; and 2. That, in investigations into the internal motions to which the bodies are subject, we need not look upon heat as a particular force, but rather as the result of the existing attractions and repulsions in connexion with certain quantities of motion imparted to the particles of the body. As to the condition in which the material particles of a body are, it is most natural to admit, with Davy, that, according to their nature, the smallest elementary particles of the body possess a certain electric force, through which they attract or repel the other material particles of the body. The proportions between the quantities of the various elements contained in the body, as well as the number of the different elements and the quantity of their electric forces, determine the positions of equilibrium of the individual particles as depending upon the adjacent particles and their internal groupings in bodies. About these positions of equilibrium, which for each individual particle are determined by the attractions and repulsions of all the other particles, the particles attracted and repelled continually vibrate on account of the imparted momentum; and, in my opinion, the heat of the body consists in this motion, which, like any other kind of motion, may be more or less, according to circumstances. Thus it is obvious that the internal quantity of energy in a body will increase as well when a new quantity of energy, whether in the shape of mechanical energy, or of heat, electricity, &c., is imparted from one body to another, as when those forces are increased by which the particles of the body are moved among themselves. On the contrary, the quantity of energy will decrease when some part of the motion contained in the body is conducted into other bodies, or when the forces decrease by which the particles of the body are moved. When no energy is imparted to or taken away from the body, and the forces by which the particles of the body are moved among themselves do not change, then the energy contained in the body will remain always the same. Now the problem is to determine the proper mathematical expressions for the energy contained in a body. In consequence of the preceding remarks this will not be difficult, as we have seen that the various kinds of energy are in reality not different, but that all of them may be considered as one-for example, as mechanical energy. As we consequently have to determine the general mathematical expression for the mechanical energy among the material particles, or, which is the same thing, to determine the mathematical expression for the total integral of vis viva which has been called forth among these particles by an originally existing source of motion, the following well-known example may be useful. If a quantity of water m be at rest at the height above the surface of the carth, and if h is small enough for us to suppose the force of gravity at the height h to be equal to the force of gravity g at the surface of the earth, then it is a truth admitted by all, and completely proved, that the whole integral of motion which may be produced and imparted (for example, to a waterwheel, or to any other machine) through the force of gravity will be expressed by Q = m.g.h; which effect, however, we shall only be able more and more nearly to approach, never to obtain entirely, on account of the impediments which always occur, such as resistance of air, resistance of friction, &c. As m.g is the weight of the water, and his the height through which the water is allowed to fall, we perceive that if m.g is expressed in pounds and h in feet, then the mechanical energy, which in mechanics generally is called the quantity of work, is to be expressed in foot-pounds-that is say, in pounds raised one foot. Further, it is well known and proved that, if we abstract from all those resistances which in fact will occur, then we obtain exactly the same quantity of work, whether the water moves vertically in the direction of gravity or is forced to move along any inclined plane or any curved line through the height h; the consequence of which is that the increment of quantity of work dQ which is developed by the falling through each little part ds of the path s is equal to m.g multiplied by ds, resolved in the direction of the force; that is, But it may easily be perceived that this formula would hold true in general, even if the accelerating force g were any variable quantity g' and m any mass, as g' will always remain constant during the element of time dt in which the element of path ds is described. If, then, we put the accelerating force resolved in the and the velocity in the path, after the lapse of time t, is put cqual to v, then we have the increment of energy expressed in general by dQ=m.4.ds=m. $.vdt. We also perceive that the unit of this quantity Q is still the same as observed before, viz. one pound raised one foot. But if the rectangular coordinates of the material point are denoted by x, y, z, and the accelerating forces in the direction. of the three coordinate axes by X, Y, Z, then we have from which the energy produced during the time t is found, viz. Q=mS (Xdx+Ydy+Zd≈) + C1, C, being an arbitrary constant. If, on the contrary, the point is not perfectly free, but subject to any material resistance, such as the resistance of a fluid, resistance of friction, &c., then the increment of energy during the time dt will only be dt2 ds dt ds di2 ds it. dt, dt from which we find the energy, which in fact is contained in the point after the lapse of time t, viz. where C is an arbitrary constant. The measure for this energy is still, as before, 1 pound raised 1 foot, which is casily ascertained by observing that the quantity of energy w might also have been obtained by causing the mass m to fall through a height h of vacuum so great that the terminal velocity thereby had been v, which depth of fall is determined from the equation |