CHAPTER I. FIRST MAIN PRINCIPLE OF THE MECHANICAL THEORY OF HEAT, OR PRINCIPLE OF THE EQUIVALENCE OF HEAT AND WORK. § 1. Nature of Heat. Until recently it was the generally accepted view that Heat was a special substance, which was present in bodies in greater or less quantity, and which produced thereby their higher or lower temperature; which was also sent forth from bodies, and in that case passed with immense speed through empty space and through such cavities as ponderable bodies contain, in the form of what is called radiant heat. In later days has arisen the other view that Heat is in reality a mode of motion. According to this view, the heat found in bodies and determining their temperature is treated as being a motion of their ponderable atoms, in which motion the ether existing within the bodies may also participate; and radiant heat is looked upon as an undulatory motion propagated in that ether. It is not proposed here to set forth the facts, experiments, and inferences, through which men have been brought to this altered view on the subject; this would entail a reference here to much which may be better described in its own place during the course of the book. The conformity with experience of the results deduced from this new theory will probably serve better than anything else to establish the foundations of the theory itself. We will therefore start with the assumption that Heat consists in a motion of the ultimate particles of bodies and of ether, and that the quantity of heat is a measure of the Vis Viva of this motion. The nature of this motion we is somewhat long, and the author has ventured to propose in its place the term "Ergal." Among the cases in which the force acting on a point has an Ergal, the most prominent is that in which the force originates in attractions or repulsions, exerted on the moving point from fixed points, and the value of which depends only on the distance; in other words the case in which the force may be classed as a central force. Let us take as centre of force a fixed point π, with co-ordinates §, n, 5, and let p be its distance from the moving point p, so that 2 p = √ (§ − x)2 + (n − y)2 + (5 — 2)*. · (13). Let us express the force which π exerts on p by p' (p), in which a positive value of the function expresses attraction, and a negative value repulsion; we then have for the components of the force the expressions X=$'(p)2=*; Y=p′(p)"=Y; Z=¢(p)5=2. ρ n-y But by (13) d = -—~: hence X=-4' (p), and simi dx == larly for the other two axes. If (p) be a function such But, since in the expression for p given in equation (13) the quantities x, y, z are the only variables, and (p) may therefore be treated as a function of those three quantities, the expression in brackets forms a perfect differential, and we may write : Xd + Ydy+Zdz == d (p) ............(16). The element of work is thus given by the negative differential of (p); whence it follows that (p) is in this case the Ergal. 19 Again, instead of a single fixed point, we may have any number of fixed points π,, π,,,, &c., the distances of which from p are P1, P2, P3, &c., and which exert on it forces $' (P1), ''(P2), ' ́(Pa), &c. Then if, as in equation (14), we assume (p), (p), (p), &c. to be the integrals of the above functions, we obtain, exactly as in equation (15), § 7. General Extension of the foregoing. Hitherto we have only considered a single moving point; we will now extend the method to comprise a system composed of any number of moving points, which are in part acted on by external forces, and in part act mutually on each other. If this whole system makes an indefinitely small movement, the forces acting on any one point, which forces we may conceive as combined into a single resultant, will perform a quantity of work which may be represented by the expression (Xdx + Ydy + Zdz). Hence the sum of all the work done by all the forces acting in the system may be represented by an expression of the form Σ (Xḍx + Ydy + Zdz), in which the summation extends to all the moving points. This complex expression, like the simpler one treated above, may have under certain circumstances the important peculiarity that it is the complete differential of some function of the co-ordinates of all the moving points; in which case we call this function, taken negatively, the Ergal of the whole system. It follows from this that in a finite movement of the system the total work done is simply equal to the difference between the initial and final values of the Ergal; and therefore (assuming that the function which represents the Ergal is such as to have only one value for one position of the points) the work done is completely determined by the initial and final positions of the points, without its being needful to know the paths, by which these have moved from one position to the other. This state of things, which, it is obvious, simplifies greatly the determination of the work done, occurs when all the forces acting in the system are central forces, which either act upon the moving points from fixed points, or are actions between the moving points themselves. First, as regards central forces acting from fixed points, we have already discussed their effect for a single moving point; and this discussion will extend also to the motion of the whole system of points, since the quantity of work done in the motion of a number of points is simply equal to the sum of the quantities of work done in the motion of each several point. We can therefore express the part of the Ergal relating to the action of the fixed points, as before, by (p), if we only give such an extension to the summation, that it shall comprise not only as many terms as there are fixed points, but as many terms as there are combinations of one fixed and one moving point. Next as regards the forces acting between the moving points themselves, we will for the present consider only two points p and p', with co-ordinates x, y, z, and x', y', z', respectively. If r be the distance between these points, we have r = √(x' − x)* + (y' − y)2 + (z' — z)* ......................... (19). ......... We may denote the force which the points exert on each other by f'(r), a positive value being used for attraction, and a negative for repulsion. Then the components of the force which the point p exerts in this mutual action are and the components of the opposite force exerted by p' are so that the components of force in the direction of x may also be written Similarly the components in the direction of y may be |