ENERGY 207 without changing its fixed point of support, all the energy of oscillation is kinetic. At intermediate positions the energy is partly kinetic and partly potential. Kinetic energy is possessed by a system of two or more- bodies in virtue of the relative motion of its parts. Since our conception of velocity is essentially relative, and we know nothing about absolute velocities in space, it is plain that any property possessed by a body in virtue of its motion can be possessed by it only in relation to those bodies with respect to which it is moving, and thus a single rigid body can never be said to possess kinetic energy in virtue of the motion of its centre of mass. If a body whose mass is m grammes be moving with a velocity of v centimetres per second relative to the earth, the kinetic energy possessed by the system is mv i ergs if m be small relative to the earth. But if we consider two bodies each of mass m and one of them moving with velocity v rela tive to the other, we can only obtain ^mv- units of work from this system alone, and we ought not to say that the system considered by itself possesses more than fynv 2 units of energy. If we include the earth in our system the whole energy will depend on the velocities of the bodies relative to the earth, and not simply on their velocities relative to one another. Hence whenever we say that the kinetic energy of a body is ^mv 2 , we mean its kinetic energy relative to the earth, and the statement is only true when the mass of the body is very small compared with that of the earth. Any general expression for the energy of a system ought to be true whatever body in the system we consider fixed. It is manifest that the expression ^mv- will not be a true re presentation of the kinetic energy of the earth and a cannon shot if we choose to consider the shot fixed and the earth moving towards it. In fact any general expression for the energy of a system must involve the masses of all the bodies concerned ; but if the mass of one body be infinite compared with that of any of the others we may adopt the expression i2(mv 2 ) for the kinetic energy, the body of infinite mass being supposed at rest. It is only when a body possesses no motion of rotation that we may speak of its velocity as a whole. If a body be rotating about an axis, it follows from D Alembert s principle that the work it is capable of" doing while being brought to rest is the same as if each particle were perfectly free and moving with the velocity which it actually possesses. Hence if the moment of inertia of a body about its axis of rotation be represented by I, and its angular velocity by to, the work which can be done by it if we can succeed in bringing it to rest will be ^Iw 2 . We shall ses hereafter how this energy may be transformed without the help of any external body if we suppose the rotating body indefinitely extensible in any direction at right angles to the axis of rotation, so that there is a sense in which we may speak of the kinetic energy of rotation as really belonging to the rotating body. When the stresses acting between the parts of a system depend only on the relative positions of those parts, the sum of the kinetic energy and potential energy of the system is always the same, provided the system be not acted upon by anything without it. Such a system is called conserva tive, and is well illustrated by the swinging pendulum above referred to. But there are some stresses the direction of whose action depends on that of the relative motion of the visible bodies between which they appear to act, while there are others whose magnitude also depends on the relative velocities of the bodies. When work is done against these forces no equivalent of potential energy is produced, at least in the form in which we have been accustomed to recognize it, for if the motion of the system be reversed the forces will be also reversed and will still oppose the motion. It was long believed that work done against such forces was lost, and it was not till the present century that the energy thus transformed was traced, and the principle of conservation of energy established on a sound physical basis. The principle of the Conservation of Energy has been stated by Professor Clerk Maxwell as follows : " The total energy of any body or system r,f bodies is a quantity which can neither be increased nor diminished by any mutual action of those bodies, though it may be trans formed into any one of the forms of which energy is sus- ceptible. Hence it follows that, if a system be unaffected by any agent external to itself, the whole amount of energy pos sessed by it will be constant, and independent of the mutual action of its parts. If work be done upon a system or energy communicated to it from without, the energy of the system will be increased by the equivalent of the work so done or by the energy so communicated ; while if the system be allowed to do work upon external bodies or in any way to communicate energy to them, the energy of the system will be diminished by the equivalent of the work so done or energy so communicated. In order to establish this principle it might at first sight appear necessary to make direct measurements of energy in all the forms in which it can possibly present itself. But there is one form of energy which can be readily measured, and to which all other forms can be easily reduced, viz., heat. If then we transform a quantity of energy from the form in which it is possessed by the earth and a raised weight, and which can be at once determined in foot pounds or ergs, into heat, and measure the amount of heat so produced, and if subsequently we allow an equal amount of energy to undergo various intermediate trans formations, but to be finally reduced to heat, and if we find that under all conditions the amount of heat is the same, and in different sets of experiments proportional only to the amount of energy with which we started, we shall be justified in asserting that no energy has been lost or gained during the transformations. It is the experimental proof of this which Joule has given us during the last thirty years, but we shall refer more at length to his work shortly. It has been recently pointed out by Thomson and Tait (Natural Philosophy, arts. 262 sqq.) that Newton was acquainted with the principle of the conservation of energy, so far as it belongs purely to mechanics. But what became of the work done against friction and such non-conserva tive forces was entirely unknown to Newton, and for long after his time this work was supposed to be lost. There were, however, some, even before Newton s time, who had more than a suspicion that heat was a form of energy. Bacon expressed his conviction that heat consists of a kind of motion or " brisk agitation " of the particles of matter. In the Novum Organum, after giving a long list of the sources of heat, some of which may fairly be adduced in support of his opinion, he says, " From these examples, taken collectively as well as singly, the nature whose limit is heat appears to be motion." In the following quotation Bacon appears to rise to the most complete appreciation of the dynamical nature of heat, nor do the most recent advances in science enable us to go much further. | It must not be thought that heat generates motion or motion heat (though in some respects this is true), but the very essence of heat, or the substantial self of heat, is motion and nothing else." Although Bacon s essay contains much sound reasoning, and many observations and experiments are cited which afford very strong evidence in favour of the theory he maintains, yet these are interspersed with so many false analogies, and such confusion between heat and the acrid or irritant properties of bodies, that we must re
serve for those who came after him the credit of having