Jan. 2, 1879] 



NATURE 



^95 



flow may make any angle within certain limits with that of the 

 momentum exchanged. If we are to adhere strictly to the ordi- 

 nary conventions with regard to the directions of forces, it is 

 clear that we would need to consider the transference of momen- 

 tum which we term force, only with reference to the direction of 

 the momentum transferred, and without any reference to the 

 direction of transference. It i?, however, evident that this 

 latter direction is of very great importance in physical investi- 

 gation, and it is a matter worthy of serious consideration whether 

 or not force should not be considered a two-directional quantity, 

 one into whose definition two directions enter. Impact is a 

 difficult subject, perhaps, just because of the large possible 

 variation of the difference of these two directions. All other 

 forces (rates of transference of momentum), except those in- 

 volved in impact may be divided into three simple classes corre- 

 sponding to compression, tension, and shear. 



In considering stress, the phrases ' ' transmission of momen- 

 tum " and "rate of transmission of momentum" areas conve- 

 nient, perhaps, as the corresponding phrases with " transference " 

 substituted for "transmission." 



The most obvious objection to this definition of force is 

 that a force may be applied to a body, and yet it receives 

 no momentum. The objector would probably say that though 

 the force be applied, yet there may be no momentum trans- 

 ferred to the body. But this would be quite WTong, as can 

 be most easily recognised if Prevost's theory of exchanges of 

 heat by radiation and the similar theory for conduction of 

 heat be recalled to mind. A body may quite easily have 

 simiJtaneously equal amounts of opposite momentum trans- 

 ferred to it. These will balance, and its centre of gravity will 

 suffer no acceleration of velocity. This remark will make it 

 evident that the theory of force gives an easy and unhesitating 

 answer to the much-debated question as to whether there are 

 really such things as unbalanced forces. A transference of 

 momentum between two bodies may just as readily be un- 

 balanced as balanced. Let us consider this balancing of trans- 

 ferences of momentum more particularly. Let a body have 

 momentum transferred to it by the pressure of another body 

 upon a certain portion of its surface. This can be balanced 

 in different ways. It may be balanced by a perpendicular 

 pull applied to a portion of the surface parallel to that 

 to which the pressure is applied, and facing the same way, i.e., 

 on the same side of the body. The directions of the momenta 

 transferred at these two surfaces are the same, but the directions 

 of transference or flow are opposite. Or the pressure may be 

 balanced by an oppositely directed pressure upon a parallel sur- 

 face facing the other way. In this case the directions of the 

 momenta transferred at the two surfaces are again the same, 

 while the directions of flow areaLso the same. In all cases when 

 a body is kept in balance by transferences of momentum going on 

 through its different surfaces, it is evident that for any amount 

 of momentum of a given direction transferred into it at one 

 surface an equal amount of momentum of the same direction 

 must be transferred out of it at some other surface. The 

 directions of transference or flow at these two surfaces may be 

 relatively any whatever — they are quite independent. The 

 balance of the body, with respect to the velocity of its centre 

 of inertia, is quite uninfluenced by the directions of the momen- 

 tum-flows through its different surfaces. But evidently the state 

 of stress and strain throughout the inferior of the body depends 

 a great deal upon the relative directions of these flows as 

 well as upon the relative positions of the surfaces. 



But, as regards the direction of the momentum, it must be 

 remembered that this depends upon what we arbitrarily choose 

 to be our standard positive direction, whereas the equilibrium 

 of the body acted on certainly does not depend in the least upon 

 that arbitrarily chosen direction. Thus, as in the above ex- 

 ample, let the body 2 be kept in equilibrium by the equal and 

 opposite pressures of the bodies I and 3 on its opposite faces. 

 The question is whether momentum is being transferred from I 

 to 2 and from 2 to 3, the momentum transferred having also 

 this direction ; or whether both the flow and the momentum 

 flowing have exactly the opposite direction, viz., from 3 through 

 2 to I. If we have a standard positive direction to go by, and 

 if I is not in equilibrium, but is being stopped in its motion by 

 impact on 2, then the above question is easily answered at once. 

 But if I is in equilibrium as well as 2, we must, in order to 

 answer the question, look beyond i to find out the direction of 

 the other transference of momentum, which, along \vith that 

 between i and 2, keeps I in equilibrium. If this other trans- 



ference takes place between i and another body which b again 

 in equilibrium, it would be necessary to go back still another 

 step in order to find out in which direction the flow is really 

 taking place. If the whole system of which these bodies form 

 parts is everywhere in equilibrium, i.e., all the parts at rest re- 

 latively to each other, we would in this way travel from one 

 body to another in a complete circuit in search of some point 

 which would disclose the real direction of flow, but without 

 ever coming to any such point. Because, following round the 

 circuit, we would again come back to 3 and 2 and i. The 

 choice of a standard direction as the positive one does not help 

 us in the least to come to even a formal conclusion. We 

 remain, however, sure of two things — first, that there is really 

 a continual flow of momentum taking place all round this 

 circuit in the system ; and, secondly, that the direction of 

 this flow is at some places, which we can definitely specify, 

 in the same direction as the momentum transferred, and at some 

 other places, equally easily specified, in the opposite direction. 

 Take as an example a piano. We may suppose the upper hori- 

 zontal bar of the frame to which the strings are attached to be 

 continually losing upward momentum, which is being continually 

 received by the top parts of the tightened strings. This upward 

 momentum the strings are continually transmitting downwards 

 from particle to particle, and at the foot of the strings it is deli- 

 vered to the bottom horizontal bar of the frame. This bottom 

 bar transmits the upward momentum horizontally, each section 

 being in shear, to the vertical sides of the frame. The trans- 

 mission down the wire is in the direction opposite to that of the 

 momentum transmitted ; in the horizontal bottom bar the direc 

 tion of transmission is perpendicular to that of the momentum , 

 through the sides this momentum flows upwards, that is, in the 

 same direction as the momentum itself, and finally, it is trans- 

 mitted again horizontally through the upper bar to be redelivered 

 to the strings. This explanation is completely satisfactory in 

 accounting for the conditions of strain of the various parts of 

 the piano. But to explain there conditions an equally satisfac- 

 tory hypothesis would be that a stream of downward momentum 

 is continually circulating through the piano in the same circuit 

 as the above, but in the opposite direction round that circuit. 

 Or again we might suppose two opposite circulations to be con- 

 tinually going on, one of upward momentum and the other 

 of downward momentum. But whichever of the three hypo- 

 theses we may adopt, we always have the flow in the string 

 which is in tension opposite to the momentum flowing through 

 it, and the flow through the horizontal bar perpendicular to the 

 momentum, and the flow through the sides of the frames in the 

 same direction as the momentum transmitted. 



Which of the three is to be chosen, or is it of any consequence 

 that we should know which should be taken ? The question is 

 not one that can be made to have any degree of unreality in 

 appearance by merely measuring the motions relatively to one 

 thing or another. It is not whether the momentum transferred 

 is upward or downward relatively to the centre of the earth, or 

 relatively to the sun or to the stars. It is, what is the direction 

 of this momentum relatively to the centre of inertia of the piano 

 frame itself, whether this relative momentum is directed from 

 one end of the piano towards the ether or from that latter to the 

 former, and the answer to this question is quite independent of 

 what we arbitrarily choose to call the absolute velocity of the 

 centre of inertia of the whole structure. 



I will venture to say that the correct answer is that there are 

 two opposite streams of equal amounts through the structure. 

 What is meant by equal amounts is, of coune, that the opposite 

 rates of transference through any section are numerically equal. 

 The simplest and clearest proof is this very simple one : If there 

 were only one stream circulating in one direction, since from 

 the above it is clear that the momentum flowing along in this 

 stream is at every point of it of the same direction, and since the 

 stream is a continuous steady one, every part of the structure 

 through which this stream flows would have the velocity corre- 

 sponding to this momentum, and in consequence the centre of 

 inertia of the structure would have a certain velocity in the same 

 direction. The inconsistency of this result with the datum from 

 which we started, namely, that the momentum transmitted was 

 to be measured relatively to the centre of inertia, need not be 

 pointed out. To look at the question in another way, let us only 

 consider what this mom.entum, this thing that is being trans- 

 ferred from particle to particle, really is, viz., mass multiplied by 

 velocity, and we cannot fail to come in a moment to the conclu- 

 sion that the£e streams are simply streams of molecular vibration. 



