Hypotheses of Dynamics. 237 



to this time-scale, in the case of motions extending over long- 

 periods of time. The first law, when expressed by reference 

 to the fixed stars and the earth's rotation, is therefore no 

 longer regarded as sufficiently accurate for all purposes ; and 

 the exact expression of the law, as empirically determined 

 and employed in actual work, changes from day to day or at 

 least from decade to decade. The question therefore arises : 

 Can we put the laws of motion into general forms such that 

 the empirical forms which they may have at any time may 

 be regarded as special cases determined by the state of know- 

 ledge of the time ? 



The latter of the two methods'* referred to above is intended 



* Mach, though holding, as seen above, to the empirical result of the 

 historical-critical method, gives in his Mechanik (p. 218) an interesting 

 " remnant," as he calls it, of his efforts to apply the second method. He 

 holds that, in using the first law in its Newtonian form, we may be 

 regarded as employing the universe, or a sufficiently large portion of it, 

 as our reference system, and on the following grounds : — " Instead of 

 saying the velocity of a mass /x remains constant in space we may also 

 employ the expression, the mean acceleration of the mass ju, relatively to 

 the masses m, m', &c, at the distances r, r', &c, is zero, or 



d 2 Hmr _q 

 ~d? 2m ~ 



The latter expression is equivalent to the former, provided we take into 

 consideration a sufficient number of sufficiently distant and great masses, 

 the mutual influence of the nearer small masses being in that case 

 negligible." If this be so, the first law may be expressed as follows: — 

 The mean acceleration of any particle, relatively to the other particles of 

 the universe, or of a sufficient portion of the universe, is zero, provided 

 the particle is not acted upon by force, — an expression which obviously 

 has not the same vagueness as the Newtonian form of the law, though 

 practically, as Mach points out, it is not more readily applicable, on 

 account of the impossibility of making the summation necessary for the 

 determination of the mean acceleration. 



How this result is arrived at, Mach does not say. But it is easy to 

 prove it to be one of the properties of the centre of mass, that the com- 

 ponent acceleration, in any direction, of any one particle of a system, 

 relatively to the centre of mass of the system, is equal to the mean com- 

 ponent acceleration of this particle, in the same direction, relatively to 

 all the other particles of the system, provided the mass of the particle is 

 small compared with the mass of the system. In making the above 

 statement, therefore, Mach would seem to assume that the uniform 

 velocity contemplated in Newton's form of the first law is a velocity 

 which is uniform relatively to the centre of mass of the universe, or of a 

 sufficiently large portion of it ; and if that be so, he assumes a partial 

 specification of a dynamical reference system. It would also appear that 

 the portion of the universe taken into consideration need not consist of 

 numerous and distant particles, but must simply have sufficient mass. 



It is obvious that if the above assumption be made, not only may the 

 first law be thrown into the above form, but also the second law may be 

 thrown into a corresponding form. The making of this assumption, how- 

 ever, introduces a complication. If we assume merely that there are axes 



