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NATURE 



[jmt. 2, iS;9 



notion that the invisible order of things which continuity requires 

 a> antecedent to the visible order, is in any sense materia!. 

 They only assume that it must be conditioned. Indeed, the 

 authors of the " Unseen Universe " have expressed this convic- 

 tion in the preface to the second edition of their work, in italics, and 

 in language that is not only exceedingly clear, but also extremely 

 strong. 



But it seems to be taken for granted on all sides that a man 

 of science can only imagine a mechanical unseen. This is really 

 very hard. 



The analogy (however inadequate) furnished by Thomson's 

 vortex atoms, and the invisible fluid which they postulate, is too 

 good an illustration of a novel and difficult conception to be 

 disregarded ; but it will have to be laid aside at once, if it can be 

 shown to be necessarily productive of such extraordinary mis- 

 conceptions even in intellects of the highest order, 



Hermann Stoffkraft 



Schloss Ehrenberg, Baden, December 25, 1878 



Force and Energy 



Since a year or two back, when Herbert Spencer started, in 

 the columns of Nature, a discussion as to the real meaning of 

 the word " force," most careful-thinking students of mechanics 

 have probably come to the conclusion that either the ute of the 

 word " force " must be discontinued as a physical scientific 

 term, or that it must be defined in a different manner from that 

 adopted almost universally by those "doctors" whose writings 

 seemed to weigh so heavily on the brain of " poor Publius." 

 Tiiey all agree in saying that in its physical application the word 

 "force" means that which produces i.e., the cause of change 

 of momentum. It is needless to give quotations. They are all, 

 except one, curiously explicit. Germans, French, and English 

 agree. "So sehen wir diese Aenderung als Wirkung irgend 

 einer in demselben thatigen Ursache an ; diese Ursache nennen 

 wir Kraft." (Ritter's "Mechanik," p. 36). "On donne, en 

 general, le nom de force a la cause quelconque qui met un 

 corps en mouvement, ou seulement qui tend a le mouvoir." 

 (Poisson: " Traite de Mechanique," Introduction, p. 2.) Al- 

 though in Tail's ' ' Recent Advances " we find on p. II " that we 

 have not yet quite cast off that tendency to so-called metaphysics 

 which has so often blasted," &c., &c. ; yet on p. 16 of the same 

 book there is reproduced the fine old crusty Newtonian maxim 

 to which Thomson and Tait and Tait and Steele cling with s^uch 

 fond reverence: "force is any cause which," &c. Clerk 

 Maxwell gives no formal definition of force in his " Electricity 

 and Magnetism." On p. 5 he simply gives its dimensions. On 

 p. 83 of his invaluable " Theory of Heat" he defines, "force 

 is whatever changes or tends to change," &c. This is a very 

 ingenious mode of escaping the difficulty by simply giving no 

 definition at all. We are told what the result of force is, but 

 not what force itself is. We are told that force is "whatever," 

 which is not very clear, Jeames would hardly think that justice 

 was done him if we asserted that the complete definition of him 

 was " whatever opens a door," and made no mention of the 

 fact of his humanity or of his grand plush breeches. It is, in 

 fact, a confusion between a statement of the mode of measuring 

 quantitatively the force, and the definition of the force itself. 

 A physical definition should certainly show clearly what the 

 proper way of measuring the quantity is ; but this latter is not 

 the definition itself. Moreover, there may be different almost 

 equally good modes of measurement, all leading to the same 

 numerical result. Clerk Maxwell's definition is clear as to a 

 mode of measuring force, but furnishes absolutely no informa- 

 tion as to the nature of the thing intended to be defined. It, 

 therefore, differs from the others in that they are real meta- 

 physical definitions, presuooably comprehensible to those who 

 understand metaphysics, while his is no definition at all. Prof. 

 John Perry, in his book on " Steam," adopts the same device as 

 Prof. Clerk Maxwell, substituting the word "anything" for 

 "whatever." Rankine forms a remarkable exception. He 

 says that " force is an action between two bodies either causing 

 or tending to cause change in their relative rest or motion," 

 Here the word "cause" is used in such a sound, practical, 

 common-sense w^y that no one could take exception to such use 

 of it, even in a physical definition, and probably "action," as 

 here used, might be explained clearly enough for all useful pur- 

 poses as " a changing relation" or "a change of relation." 

 Kankine, however, does not take the trouble to do this last. 

 Now clearly a cause is a metaphysical entity, if it is an entity 



at all, and from the very nature of the difference between meta- 

 physics and physics, a metaphysical entity cannot possibly be 

 made of any use in physical investigations. If, then, the word 

 force is to be usefully employed in physics, it must be defined 

 as something else than a "cause," When we talk of forces, 

 the physical facts the observation of which we think of, are 

 accelerations of momentum ; and in his Glasgow lecture Prof. 

 Tait seems half inclined to use "force" and "acceleration ot 

 momentum " as synonymous terms. But an acceleration of 

 momentum is a function of one body only ; and every one knows 

 that what is mentioned in Rankine's definition is true, namely, 

 that force is a function of two bodies, and can have neither 

 objective nor any other kind of existence except as a relation 

 between two bodies. Seeing that it is so, I beg to lay before 

 your readers for their favourable consideration the meaning of 

 the word force which I have used for several years past, 

 I wish force to be defined as "time rate of transference of 

 momentum." A transference of anything can only take place 

 between one body and another, and in the transference the 

 amount transferred from the first body to the second is neces- 

 sarily equal to the amount transferred to the second body from 

 the first. This might seem to be such a truism as to be a mere 

 repetition of words ; but we must remember that it is the law 

 of motion which the " transcendently lucent" Newton dis- 

 covered from his extensive physical experience ; and, in order 

 to discountenance scepticism, we might add, by-way of paren- 

 thesis, that during the transference no spilling takes place. 



" Poor Publius" might thus get a hint that there is such a 

 physical fact as conservation of momentum which is independent 

 of all formal definitions. If momentum is conserved, i.e., if it 

 has an enduring existence so that at one time there is no more 

 nor less of it than at another, then during a direct transference 

 of some of it from one part of the system in which it is lodged 

 to another part, the amount lost by the one part must evidently 

 be the same as that gained by the other part. Thus an accele- 

 ration or time-rate of gain of momentum to one part necessarily 

 implies a simultaneous equal time-rate of loss of momen- 

 tum from another part, and also a simultaneous equal rate of 

 transference of momentum from that other to the first part. All 

 these three rates have directions inasmuch as they are time-rates 

 of directed quantities. The first is a rate of gain of momentum, 

 which momentum has a certain direction. If that direction be 

 reckoned positive the gain is one of positive momentum, and 

 the acceleration is naturally reckoned as positive. The second is 

 a rate of loss of momentum of the same direction, i.e., a loss of 

 positive momentum which is equivalent to a gain of negative 

 momentum, and therefore this time-rate is naturally reckoned 

 negative. The meaning of this is simply that the proper physical 

 sign to ascribe to acceleration of momentum is the directional 

 sign of the momentumgained. The two opposite signs of the above 

 two rates have given rise to the idea of two equal and opposite 

 forces acting between the bodies. If the forces were located in 

 the bodies and not between them, the phraseology would be 

 consistent with Tait's definition of force as simply "accele- 

 ration of momentum." But I do not hesitate to say that this 

 idea of force is quite unnecessarily out of accord with the com- 

 monly received notion of force as a mutual action or relation 

 between two bodies, because in this view force would distinctly 

 have reference to only one body. If, however, we use force 

 to mean the transference of momentum, there is, of course, only 

 one force between the two bodies. The question is what sign is 

 to be given to this force, and it is not quite easy to answer. 

 Force is in this view a flux, a rate of flow of momentum. This 

 flow takes place in a certain direction, and it is the flow of a 

 directed quantity. Are we to take the direction of the flow or 

 the direction of the momentum that flows, to determine the 

 proper sign of the force ? These two directions need not be the 

 same. Thus in a bar subjected to tension the flow of momentum 

 is in the direction opposite to that of the momentum itself. In 

 a bar in compression the flow of momentum takes place in the 

 same direction as that of the momentum. In a mass subjected 

 to shearing stress the direction of the flow is perpendicular to 

 that of the momentum. In the case of the attraction of gravi- 

 tation between two bodies the direction of the flow of momentum 

 is always the exact opposite of that of the momentum that flows 

 from one to the other in whatever way the two may be moving. 

 In the case of impact if we take the direction of the flow of mo- 

 mentum as that of the perpendicular to the surfaces that touch 

 during impact drawn from the bod\ that loses momentum 

 towards the body that gains momentum, then this durection of 



