f5o 



SCIENCR 



[Vol. XX. No. 501 



hence the two assertions about Force are arbitrary and may be 

 laid down as a (partial) definition of that term. 



From this definition all the theorems of dynamics may be de- 

 duced, as from Newton's laws of motion. The theorems of statics 

 may also be deduced, the only difHculty being the principle of 

 virtual work. This difficulty, however, disappears as soon as the 

 term "geometrical conditions" is properly defined. 



We have then a symbolic dynamics. To give it a subjective 

 meaning we have to conceive a real denotation for its terms. It 

 is not, however, necessary to give a real denotation to Force if we 

 can do so to Mass, for we may still regard Force as merely a name 

 for the product of mass by acceleration, or (which is the same 

 thing) as the time-flux of momentum. To give the theory an 

 objective application it is necessary to show that what we call 

 material particles not only occupy positions which are continuous 

 one-valued functions of what we call Time, but also possess a 

 certain characteristic which is not a function of space or time, and 

 which may be called Mass. Then, whether we attach any deno- 

 tative meaning to Force or not, we can discuss the forces or 

 stresses that must be postulated between various particles of 

 matter. The magnitudes of these will in general depend on the 

 axes we assume by which to determine positions, and also on the 

 masses assigned to the various particles. The axes and masses 

 are therefore assumed in such a way as to make the resulting 

 system of stresses the simplest possible. For example, it is gen- 

 erally assumed that the stress between any two particles dimin- 

 ishes as the distance between them increases, and may be neglected 

 if this distance is very great. Hence in astronomy the attractions 

 of the fixed stars on the planets may generally be neglected, and 

 we may discuss the solar system alone. It is further shown that 

 the system of stresses between the sun and planets is simplest 

 when a certain plane is taken as " the invariable plane." But we 

 do not really knoiv that the stresses thus deduced are the actual 

 ones, or indeed that there is any actual phenomenon correspond- 

 ing to what we call stress at all. Any plane might be chosen as 

 the "invariable" one, at the cost of having to postulate a more 

 complicated system of stresses. We cannot determine fixed di- 

 rections dynamically, any more than kinematically, except by 

 making assumptions which are reaUy arbitrary about the stresses 

 between certain particles. 



As Professor MacGregor points out, the law of the conservation 

 of mechanical energy would flow from the assumption that 

 stresses are functions of the distances between the particles on 

 which they act. But this would not include the general law of 

 conservation of energy until all energy was shown to be mechani- 

 cal energy. And even then, on the above assumption, the term 

 conservation of energy would be rather misleading; for the kinetic 

 energy is not conserved unless the term potential energy is merely 

 used as a cloak to hide our ignorance of kinetic energies which 

 for the moment have passed beyond our ken. For example, a few 

 years ago it might have been said that when we project a keeper 

 away from an electro-magnet, the kinetic energy with which it 

 starts becomes converted into potential by the time it stops, just 

 as when we throw a stone into the air. But if, while the keeper 

 is at a distance from the magnet, the current is switched off, that 

 potential energy is abolished! The true view is. however, that 

 there never was any potential energy at all, the energy of the 

 flying keeper had its equivalent in an increase in the electric cur- 

 rent round the magnet — a kinetic, not a potential, energy. And 

 I have no doubt that some day science will show a similar ex- 

 planation to hold with respect to gravitation and other actions at 

 a distance. When that day comes the term " potential energy " 

 may be banished to " the limbo of once useful things." 



It will be seen, therefore, that I differ from Professor MacGregor 

 chiefly in denying " the non-relative character of Force." Pro- 

 fessor MacGregor says, " it is easy to show that if it [the third law 

 of motion] hold for one point of reference, it cannot hold for an- 

 other having an acceleration relative to the first." I should like 

 to see his proof ; but if he refers accelerations to a single point, I 

 can well understand that he should arrive at results inconsistent 

 with mine. For, as I have shown, the apparently absolute de- 

 terminations of direction depend in reality on arbitrary assu raptions 

 as to stresses. Having made these arbitrary assumptions, it may 



well be iuipossible to further make arbitrarily the assumptions in- 

 volved in the third law of motion. 



I cannot quite follow his paragraph beginning " It may easily 

 be proved that the stress between two particles is proportional to 

 the product, by the sum of their masses into their relative accel- 

 eration." There seems to be some misprint; but how a single 

 particle could in any case exert all the forces acting on a system 

 of particles, I cannot understand, unless the words "equal and 

 opposite" in the third law of motion are not held to imply that 

 the forces act in the line joining the particles, which, moreover, 

 is distinctly implied in the Professor's law of stress. In any case 

 the difficulty referred to above comes in again, viz., that we can- 

 not determine dir.ections absolutely, or positions by reference to a 

 single point. 



In conclusion, I should like to point out that it seems incon- 

 venient, even if Professor MacGregor's views be accepted on other 

 points, to include in one law of stress, two statements resting on 

 such very different evidence as that forces may be considered to be 

 attractions or repulsions, and that their magnitudes depend solely 

 on the distances between the particles on which they act. It 

 would give a student a very false notion of the fundamental hy- 

 potheses of dynamics to teach him that he must accept or reject 

 both these assertions together. Edward T. Dixon. 



Cambridge, Eng., Aue. 20. 



The Fundamental Hypothesis of Abstract Dynamics. 



Professor Hoskins points out {Science, Aug. 26, p. 123) that 

 for the conservation of energy the necessary and sufficient con- 

 dition is that 'S Pdr shall be a perfect differential of a function 

 of the quantities r, P being the stress between any two particles 

 of the system, and r their distance; and that the condition that 

 each P shall be a function of the con-esponding r only, which I 

 suggested for adoption as a fourth law of motion, with a view to 

 the deduction of the law of the conservation of energy {Science, 

 Aug. 5, p. 74), while sufficient, is not necessary. 



There are three reasons which influence me in selecting for the 

 fourth law an hypothesis which is more than sufficient for the 

 main purpose in making the selection, viz., (1) that it is capable 

 of simple physical expression, (3) that it is already known to hold 

 in the case of several natural forces, and (3) that the additional 

 assumption involved in it, over and above that necessary for the 

 deduction of the conservation of energy, is one which is, I think, 

 invariably made in investigations on the laws of natural forces. 



What the additional assumption is, is readily seen. In a system 

 of two particles A and B, '2 Pdr becomes Pdr ; and in this case 

 it is both necessary and sufficient for the conservation of energy 

 that the single stress acting shall be a function of the distance 

 AB only. If we add a third particle, C, to the systena, conserva- 

 tion no longer requires that the stress between A and B shall be a 

 function of the distance AB only, though it is secured if that con- 

 dition is fulfilled. Thus the proposed law assumes, in addition to 

 what is required for conservation, that the stress between A and B 

 is not changed by the fact that other stresses have begun to act 

 between A and C and between B and O. The proposed law there- 

 fore involves an assumption similar to that implied in Newton's 

 second law. As Newton's law assumes that a force produces the 

 same acceleration in a particle whether other forces act on it or 

 not, so the proposed law assumes that the stress between two 

 particles is the same whether or not there are other stresses acting 

 between them and other particles. 



That this additional assumption holds in the case of somenatural 

 forces has been abundantly verified, and in investigations into 

 the laws of forces not yet determined, so far as my knowledge of 

 such investigations goes, the same assumption is always made. 

 This being so, we would seem to be warranted in adopting, tenta- 

 tively of course, as a fourth law of motion an hypothesis in which 

 this assumption is implied. The proposed law cannot be said to 

 have received anything like the verification that Newton's laws 

 have received. But of the many deductions which have been 

 made from it, none have been contradicted, while many have been 

 corroborated, by experience. J. G. MacGregor. 



Shubenacadle, N.S., Sept. 2. 



