390 



NA TURE 



[February 22. 1^94 



lively to those axes. We suppose different particles to 

 have any accelerations relative to those axes which may 

 be assigned, or which are deducible from data given, and 

 so from the configuration at any given epoch that at any 

 other, that is, to speak shortly, the motion, can be found. 

 If the particles do not change their configuration relatively 

 to one another a limitation is imposed on the motion, 

 the particles constitute a rigid body. Thus we may con- 

 sider any conceivable cases, and the science which deals 

 with them is one of pure kinematics. 



Now we may st'ppose our reference system, which we 

 may call A, to have a motion relatively to some other 

 reference system B, and the motion of the particles con- 

 sidered if referred to that other system will be com- 

 poundccl, for any instant, of the motion which the particles 

 would have with respect to B, if they were rigidly con- 

 nected with A, in the positions they have at that instant, 

 and of the motions which the particles then have with 

 respect to A. There is no difficulty, if the motion of A with 

 respect to B is specified, in determining the former part 

 of the motion of each particle. It will vary, of course, 

 with the changing positions of the particles in consequence 

 of their motions with respect to A. 



Similarly we can push the reference still further back, 

 and so from reference system to reference system when- 

 ever we find it desirable to do so. Of course we should 

 never by any such process as this reach axes absolutely 

 fixed ; but it is the process by which we introduce correc- 

 tions suggested by experience, as explained below. 



It is, then, a result of observation that we can stop at 

 some reference system, it may be the first A, which is 

 suggested to us by the circumstances of the case. To 

 a certain extent we can consider the effect of referring 

 our chosen reference system to other reference systems 

 naturally suggested, and be sure that the additional 

 motions necessary for the parts of our system are 

 negligible. 



In practice we generally make the supposition that 

 we may refer to a naturally suggested system of reference 

 and find in what manner the results deduced require 

 correction. For example, we refer the motion of a 

 projectile to axes fixed in the earth, say one vertically 

 upwards, and two others, one north the other west, and 

 consider the motion. We find that the results only 

 approximately coincide with experience, and we have to 

 correct them on account of the earth's rotation. It may 

 be that there are other corrections which on account of 

 their smallness relatively to unavoidable errors of 

 observation we can take no account of. 



So far we have made no mention of mass or inertia. 

 This idea is derived from experience of physical 

 phenomena. 



If we wish to apply our ideal science to the investiga- 

 tion of physical relations from experimental or observa- 

 tional data, we can only do so on certain assumptions 

 tacitly or explicitly made, and these are to be regarded 

 as postulates to be justified by the consistency and 

 accuracy of our results when tested in their turn by 

 observation. The term axiom, it may be remarked, 

 seems inapplicable to many of these unproved assump- 

 tions, inasmuch as though they are simple concise state- 

 ments, neither their truth nor their falsehood commends 

 itself at once to the mind. 



Now, with reference to our naturally chosen system of 

 axes, we find that different bodies have, iti the same 

 circumstances, different accelerations, and hence we get 

 the idea of the masses of bodies. In estimating similarity 

 of circumstances we assume the constancy of the 

 physical properties of materials, such as constancy of 

 the quantity of matter in a body, the elastic properties of 

 a spring, and the like. Thus, if we take a given spiral 

 spring and apply it repeatedly to the same body with the 

 same stretch, we find the same acceleration given to the 

 body each time. Of course this result might be pro- 



NO. T269, '^OL. 49] 



duced by a. pari passu variation of the mass of the body, 

 and the properties of the spring, but since we find the 

 results consistent with those obtained with different 

 masses and springs, the possibility of such variations 

 need not be discussed. To this ideal method of compar- 

 ing masses, the ordinary method by weighing is shown 

 to be equivalent by Galileo's experiment with the falling 

 bodies, Newton's pendulum experiment, &c. 



Thus applying similar circumstances (which we may 

 typify by a spring with a given stretch) to different 

 bodies, we find their accelerations different, and we are 

 led to a comparison of their masses, and, thence to a 

 prediction of the accelerations which in different cir- 

 cumstances will be produced in the same mass or in 

 different masses, that is to the comparison of rates of 

 change of momentum or of force. For example, suppose 

 a spring with a given stretch in it to be applied for a 

 second to each of a number of masses, and let the 

 accelerations produced be oj, a^, a.j, &c. Then if we take 

 quantities inversely proportional to aj, a.,, a.^, &c., say 

 /Li Oj, /i a.), M, O;;, &.C., aud multiply each of these by the 

 accelerations produced, we obtain, of course, the same 

 product fi in each case, and we take this as a measure of 

 the stress in the spring regarded as the producer of 

 motion in bodies. In the ordinary system of measuring 

 forces we take yL as ma, where m is the mass of the body 

 reckoned in terms of a chosen unit of mass. This gives 

 the dynamical method of comparing the masses of 

 bodies. The masses of the bodies here considered are 

 /aaj, iJ.la.j, &C. 



On the Other hand, when we have to compare the 

 motion-producing powers of springs having different 

 stretches, that is, the forces they exert, we may use the 

 same system of bodies if we please (or any system of 

 which the masses have been compared as just described), 

 and suppose that accelerations a\, a'.,, a'^, &c. are pro- 

 duced by different springs applied to the bodies. Thus 

 applying the method of reckoning explained above, we 

 are led to measure the forces exerted by the springs by 

 the products ixa\ja^, fiu'^a.,, &c. 



Thus from the point of view here adopted, Newton's 

 second law sets up this mode of comparing masses and 

 forces, and thereby furnishes a perfectly simple and con- 

 sistent method of writing in a form ready for solution 

 the equations of motion of a body relatively to any 

 system of axes which we know from experience we may 

 regard as at rest. 



Here I wish to remark that when we write such equa- 

 tions as 



mx = X, tny = V, ms = Z, 



the quantities on the right, commonly called the applied 

 forces on the particle of mass m, are, it seems to me, 

 merely put provisionally for values of the quantities on 

 the left, which from the given circumstances of the 

 motion, that is from the relations and data given, we may 

 be able to calculate, or to supply from the results of ex- 

 periment or observation. There is not any necessity for 

 considering them as the causes or the measures of the 

 causes of the accelerations x,y, z, of the particle. 



The idea of force as cause of acceleration is useful as 

 enabling us to speak and write with brevity about dyna- 

 mical problems, and so to arrive quickly at the necessary 

 equations. For example, take the problem of the motion 

 of a particle of mass m hung by a massless spiral spring 

 which the weight of the particle stretches by a length s. 

 Then we know (i) that the stretch of the spring if not 

 counteracted by the weight w^g'of the particle would cause 

 the particle to receive an upward acceleration g, and 

 since experiment shows that different weights stretch 

 the spring by amounts proportional to them, we infer 

 (2) that when the spring is stretched by an amount 

 s-\- X, the elastic reactiom would produce an acceleration 

 g{s-\-x)ls. Hence an upward acceleration of amount 



