2l8 



NATURE 



\Jan. 9, 1S79 



With regard now to the investigation of the equilibrium or 

 the acceleration of momentum of a body founded on a know- 

 ledge of the forces acting upon it, that is, of the various rates of 

 transference of momentum between it and other bodies through 

 its surfaces, we must evidently give signs to the various surfaces 

 of the body, surfaces which are parallel, and face opposite ways, 

 being given opposite signs. If we multiply the transferences of 

 momentum taking place through the different surfaces each by 

 the sign of the surface through which it takes place, and add all 

 these products together, the sum will be the acceleration of mo- 

 mentum of the body. Thus two equal tensive forces acting 

 through parallel opposite faces would keep the body in balance. 

 Or two numerically equal forces, the one tensive and the other 

 compressive, acting through parallel surfaces facing'the same way, 

 would also keep the body in balance. If the phrase " intensity 

 of force " be used to mean the force per unit area through any 

 surface, so that it is simply a generalisation of the two common 

 phrases "intensity of pressure" and "intensity of tension;" 

 then if each small element of the surface of a body be given its 

 proper sign and multiplied by the intensity of force through 

 that small element, this force being also given its proper sign, 

 and if all such products for the whole surface be summed up, 

 the total will be the acceleration of momentum of the body. 

 The direction of this acceleration will be shown by the sign of 

 this total, the sign having reference to the relative position of 

 the surfaces which have been arbitrarily called positive and nega- 

 tive. Thus let two tensions, i.e., positive forces, be applied to 

 two parallel opposite faces, and let the force applied to the posi- 

 tive face be greater than that applied to the negative face ; then 

 the body will suffer a positive acceleration of momentum ; that 

 is, an acceleration in the direction from the negative face 

 towards the positive face. The faces perpendicular to the 

 positive and negative faces must be given the signs -t- V — i 

 and — V - I. 



Thus a pair of positive forces applied to faces with the signs 

 -t- I and -1- V - I cannot possibly balance each other. But a 

 positive force applied to a -f face can be balanced by a 

 tangential, or shearing, force applied to a ^ — \ fa ce. Be- 

 caus e the shearing force has either the sign -f ^ — x or 

 ~ V — I, and multiplied by the sign of the face, gives either 

 •f I or — I, as the sign of the product. Surfaces oblique to 

 what is chosen as the positive direction must -be considered as 

 partly scalar and partly vector, as also forces oblique to the sur- 

 faces through which they act, or rather oblique to their direction 

 of transmission. Oblique surfaces must be multiplied by oblique 

 forces according to the ordinary rule of vector multiplication. 

 This system of notation requires no further explanation, I think, 

 to those who are likely to approve of it. 



It has become lately a common habit to look upon those things 

 which are conserved, that is, those which have an enduring 

 existence, as objectively real ; while those which may come into 

 existence and go out of it again are considered as objectively 

 unreal. Whether this is a correct philosophic habit or not, it 

 has certainly tended to create suspicion as to the objective reality 

 of all mechanical quantities. A gradually extending recognition 

 of the relativity of these quantities is apt to lead on to a reluctant 

 apprehension that all so-called physical facts are mere formal 

 logical deductions from arbitrary definitions. The dark shadow 

 of distrust first fell upon momentum because the fact that it is 

 distinctly a relative quantity is most easily recognised, and thus 

 became earlier a part and parcel of our familiar ideas. Then 

 somebody suddenly recalled to mind the distinction, according to 

 definition, between external and internal kinetic energy, and 

 found that the external kinetic energy which it had been fondly 

 hoped had some lingering flavour of the ABSOLUTE still clinging 

 to it, was no more than a part of the internal kinetic energy of a 

 larger group of bodies ; and it became clear at a glance that energy, 

 that grand absolute reality which, being once borne into 

 existence by triumphant modem science is now far too carefully 

 conserved by its enthusiastic worshippers to allow of there being 

 any risk of its dropping again out of existence, is just as purely 

 relative in its nature as the velocity which has to be squared in 

 order to calculate its amount. It had been thought that because 

 a velocity has a direction and the square of a velocity has no 

 direction, therefore we might calmly and fearlessly contemplate 

 the total or partial destruction of momentum, steadfast in the 

 assurance that energy would still live for us. And thus with 

 much waiting in fluttering hope and trembling fear upon the 

 brink of the Unseen Universe, and becoming impatient at the 



non-arrival of any clear intimations of immortality for ourselves, 

 or for energy, or even for matter itself — which is clearly neither 

 more real nor more unreal than her faithful spouse energy — a 

 cloud of dismal despair seemed to be settling on-the heads of the 

 scientific nations, when a stern but cheering voice was heard 

 from Munich bidding us be satisfied with our finite .^human 

 faculty of perceiving relations only, and promising us that, if 

 we would only not aspire to divine knowledge of the absolute, 

 we might know even now and also hereafter. 



While admitting fully the relativity of all the physical facts 

 which we may learn, I think it would be very unfortunate if we 

 were to allow ourselves to confuse this with the idea that all 

 mechanics is a mere phantasmagoria conjured up by a process 

 of formal logical deduction from a basis of abitrary defini- 

 tions. The clearest exponent of this theory of formality in 

 mechanics that has come to my notice is Dr. V. A. Julius, in his 

 letters on "Time" to Nature, vol, xvi. pp. 122, 420. The 

 argument may be thrown into the form of four short pro- 

 positions and a conclusion, all of which are derived by purely 

 formal reasoning from the ordinary definitions of the various 

 quantities involved, and which a friend of mine pretends 

 make out a "clear demonstration of the utter absurdity, futility, 

 and falsity of all mechanics," 



1, All motions and velocities are simply relative. Within a 

 given isolated system, nothing with reference to the motions 

 of its parts can be known beyond the motions of these parts 

 relatively to the centre of inertia of the system. 



2, Relatively to any other system, or single body, the velocity 

 of the centre of inertia of this first system is, by definition, 

 simply the mean of the velocities of its parts. The sum of the 

 velocities of the parts relatively to the centre of inertia of this 

 first system is, therefore, always zero. 



3. Within one portion of this system, therefore, there cannot 

 be any loss of average velocity without there being a simul- 

 taneous equal gain of average velocity in some other portion. 



4. The changes which can possibly take place in the system 

 with regard to velocity consist, therefore, in balanced exchanges 

 of relative momentum between its parts, and, therefore, the 

 equality of "action and reaction" — whether calculated with 

 reference to rate of transference of momentum, or with re- 

 ference to rate of transference of energy, i.e., rate of doing 

 work — is a purely formal deduction from the definition of the 

 centre of inertia. 



Conclusion, A purely formal deduction from an arbitrary 

 definition is just as likely not to agree with reality as to agree 

 with it. Q.E.D. 



The fallacy of the argument lies in the artful omission of a 

 few words in 3, which are necessary to make the meaning quite 

 explicit. At the end of 2, the sum set equal to zero is that 

 of the velocities relatively to the centre of inertia of the system 

 itself. These, therefore, are the velocities referred to in 3. 

 Therefore, in 4, the exchanges of momentum that are balanced 

 are those of momentum measured relatively to the centre of 

 inertia of the system itself ; and it does not at all follow, by 

 pure logic, that such a balanced exchange of momentum rela- 

 tively to this centre does not produce an acceleration of 

 velocity of the centre of inertia relatively to some body outside 

 this system. Of course, if we add this outside body to the first 

 system, then pure logic will compel the exchanges of momentum 

 throughout this new combined system and measured relatively to 

 the centre of inertia of the new combined system to balance. 

 But pure logic does not necessitate the exchange of momentum 

 within one part of the system relative to the centre of inertia of 

 that part being unaccompanied by a simultaneous exchange of 

 momentum between that part and some other part, or every 

 other part. Thus the fact of conservation of momentum is not, 

 that when two bodies exchange momentum, the amounts lost and 

 gained measured relatively to the centre of inertia of the two, are 

 numerically equal, — that would be a mere truism — but that the 

 amounts lost and gained measured relatively to a third body are 

 equal to each other. This latter is a physical fact, only to be 

 proved by experiment, not by logic. The statement that action 

 and reaction between two bodies are equal, does not mean any- 

 thing in particular ; but the statement that the action of a force 

 between two bodies does not accelerate the velocity of their 

 centre of inertia relatively to a third body is a statement of 

 experimental fact. The mechanics of a system of two bodies 

 might be built up by means of formal reasoning alone ; but not that 

 of a system of three, or of more, bodies without the experi- 

 mental establishment of the law of conservation of momentum. 



