Jafi, 1 6, 1879] 



NATURE 



243 



The amount of kinetic energy lodged in this length measured 

 relatively to the same set of bodies as v^ relates to, is 



of which there is transmitted forwards per xmit of time the 

 amoimt 



\.\ FmICz-o + if - »„'} = iP'MZ'fo + \v\ - \eE{v^ + \v\ 

 This is equal to the force multiplied by the mean velocity of the 

 material. The truth of this last proposition is quite independent 

 of which group of bodies the velocities are measured relatively 

 to. The energy transferred is to be measured relatively to the 

 same group as that to which the mean velocity of the material is 

 measured. But the expression for the rate of transference of 

 energy consists of two parts, only one of which varies with the 

 choice of axes of velocity-measurement. Thus the acting-force 

 — \eE = rate of transference of momentimi ; the mean velocity of 



the material = z'„ + ie .1^; and the rate of transference of 



energy — \eE \ v^-V \e ^/ ~ \ = rate of doing work. The 



sign of the last rate indicates simply in which direction energy is 

 flowing. The sign depends on that of the mean velocity and 

 on that of e. Here ^ is a linear strain, and must have the sign 

 -f or - . If it is a twist it should have the sign V' - i or 



The constant part of the energy transferred — that part inde- 

 pendent of the axis of reference — is \^E » /■^. This and the 



amount of the acting force cannot be altered in any way by 

 varying the choice of axes. This result at first sight seems some- 

 what contradictory to the notion that cnei^ is a thing of infinitely 

 greater objective reality than force is. The amount of the 

 momentum of a body's visible motion and the amount of its 

 energy can be made just as great or as small as we please, by 

 simply imagining one or other group of bodies to be at rest. In 

 this way its momentum may te mcde to vary at w ill from — 

 infinity to -f infinity, while its energy may be made to vary from 

 zero to + infinity. Etymologically the words "force," "mcmen- 

 tum," and "energy" are mere names, but the first, force, has 

 objective reality in the sense that it is related only to the funda- 

 mental units of mass, space, and time, and does not depend at 

 all upon an arbitrary choice of axes ; while the second and 

 third, momentum and energy, are simply products, of the imagi- 

 nation. 



The first of these statements, viz., that respecting the physical 

 reality of force in the sense above explained, may be objected 

 to because of the appearance of velocities in the expression for it 

 \eE — \Vn.v. Eut here the first velocity - V is* the length 

 of material passed through by a wave of longitudinal momentum 

 in imit time, and it is an experimental fact that this is quite 

 independent of the velocity of the material measured relatively 

 to no matter what set of axes. The second velocity v is double 

 the mean velocity of the material after the force has begun to 

 act, measured relatively to a set of axes, relatively to which the 

 material was at rest before the force began to act. Thus, z/ may 

 be looked upon as containing in itself the definition of the axes 

 relatively to which it is to be measured, and thus its magnitude is 

 not at all at the disposal of cur imagination. 



Similarly the rate of transference of energy measiured rela- 

 tively to a set of axes with respect to which the material w as at 

 rest before the energy began to be uansf erred, is absolute in the 

 sense that we cannot arbitrarily alter its magnitude by an 

 exercise of the imagination. 



\ye have in the above supposed a single stream of motion 

 flowing continuously onwards and through the material under 

 consideration, so that that material neither gained nor lost on the 

 whole momentum or energy. If the portion of material con- 

 sidered does not pass on the whole motion it receives, but retains 

 either a part or the whole of it, its rate of gain of energy is to 

 be found by applying the above equations to its one or two or 

 more surfaces, or surface layers, through which transfers of 

 energy are going on. If it is receiving energy only through one 

 surface and losing it thrctgh no surface, its rate of gain of 

 ena^ \%\eY. [z-o -f ^ z/,, where e is the strain at the receiving 

 surface, and (z-o -f \ v) the velocity of that surface, measured in 

 the proper directions. It is to be observed that new finite 

 increments of velcciiy gradually sjread ever the whole material. 



so that each small part is accelerated by fits and starts, and the 

 whole mass is accelerated by what might be called pulsations, 

 or, in the case of the strains being shearing ones, "in a sort of 

 wriggling fashion. The surface particles have at any instant 

 the velocity Vc, say. They instantaneously gain the velocity v 

 and immediately after>vards lose it again, and experience this 

 change a great many times for an interval during which their 

 time-average velocity is (z/^ -J- ^ z). After this interval they for 

 another equally long interval alternate between the velocities 

 (z/o -f v) and (z',, -1- 2 z/) in such a way that their time average 

 velocity is (z/, -f \\v). 



The gain of energy in one unit of time is in magnitude evi- 

 dently dependent on r-o : that is, on the axes of reference arbi- 

 trarily chosen. Thus, not only can we alter the magnitude of 

 the energy resident in a body arbitrarily by choosing different 

 sets of axes, but, by a simple exercise of the imagination, we 

 can set the energy posEe;?sed by any portion of the universe in- 

 creasing at any arbitrarily desired time-rate. The momentum 

 may be imagined what we like, but we cannot exercise cur 

 imagination upon its rate of transference, or force ; on the 

 other hand, both the amount at any time and the rate of trans- 

 ference of energy we may make what we please. This last, 

 however, dees not at all invalidate the conservation of energy 

 as a proposition concerning the energy mieasured relatively to a 

 given set of axes ; because, although the time-rate of gain of 

 energy of one portion of the collection of bodies investigated 

 may be increased by changing from one set of axes to another, 

 still that change creates simultaneously a correspondingly in- 

 creased rate of loss of energy in another part, namely, that other 

 part from which the energy is being transferred to the former. 



It is to be observed that this change arbitrarily accomplished 

 in the magnitude of rate of exchange of energy is only possible 

 if a force is acting. If no force is acting, e is zero, and the rate 

 of exchange of energy is zero, w hatever Vo be. 



This mathematical possibility of altering, by a change of 

 motion-axe«, the time-rate of gain of energy of any special por- 

 tion of the system, seems to me to furnish the strongest con- 

 ceivable .-.rgument in favour of there being existent no other 

 kind of energy except that of motion, i.e., kinetic energy, re- 

 presented algebraically by the formula \ MV. If the conser- 

 vation of energy is true in any sense which will include kinetic 

 energy z% part of the energy which is conserved, and if the rate 

 of transference of enei^ from one part of the system to another 

 can be altered by arbitrary changes of the velocities effected by 

 choosing different axes, then there can be no energy that is not 

 energy of relative velocity. 



Comparing the kinds of reality to be ascribed to "force and 

 to "energy," we see that while force has quantitative definite- 

 ness quite independent of the stand-point arbitrarily assumed by 

 the physical imagination in viewing them, it lads that kind of 

 reality which some believe to be an attribute of those things 

 only which are "conserved," because force comes into existence 

 and goes out of it again. This kind of reality may be more or 

 less aptly illustrated by supposing that the personality of a 

 human being be not immortal but to come into existence either 

 gradually or suddenly with the birth of the human being, and to 

 go out of existence with its death. If this were the case, and 

 if the results of the temporary existence of this human being 

 were always to live in the subsequent histcrj' of human pheno- 

 mena, then force would have very much the same sort of reality 

 as the personality of a human being. On the other hand, the 

 quantity of energy that exists depends on this standpoint of the 

 imagination, but so long as this standpoint is unchanged there 

 is no change in the amount of the energy. In other words it is 

 "conserved." So long as the poiition from which it is viewed 

 is not shifted energy can neither be created nor destroyed. To 

 make for energy an illustration somewhat parallel to the above 

 made for force, suppose that all mankind had agreed upon a 

 certain unit of goodness, and that the Deity was a thing the 

 amount of whose goodness, measured by this unit, was really 

 dependent upon the characters of the philosophies believed in by 

 different sets of men, or upon the characters of the men them- 

 selves, then the beneficence of the Deity w ould be constant so 

 long as the philosophic stand-point from which he was c«n- 

 sidered remained the same, and would have no other kind of 

 constancy. If this w ere the case then energy would have very 

 much the same kind of reality as the Dtity. Again, momentum 

 is conserved in the same way as energy. Also, force being the 

 rate of transference of momentum, the real existence of U rce 

 implies also the real existence of motion, of which energy is 



