238 Prof. J. G. MacGregor on the 



to give laws of this kind. Prof. James Thomson may be 

 said to have employed it when he showed how, by observation 

 of successive relative positions of particles given as moving 

 in straight lines, the ;ixes by reference to which their paths 

 are rectilinear may be determined geometrically*. Thomson 

 and Tait may be said to employ it also, when they show, 

 by a deduction from the first law, how we may imagine 

 ourselves as obtaining " fixed directions of reference" |« But 

 these authors make no attempt to give a formal specification 

 of a dynamical reference system. 



Lange employed this method in the paper referred to above, 

 basing his suggestion as to specification on a kinematical 

 result, viz. that for three, or fewer than three, points, which 

 are moving relatively to one another in any way whatever, it 

 is always possible to find a system of coordinates, indeed an 

 infinite number of such systems, by reference to which these 

 points will have rectilinear paths ; while for more than three 

 such points this is possible only in special circumstances. It 

 follows that the law of the uniformity of the direction of 

 motion of particles free from the action of force is, for three 

 such particles, a mere convention, and that it is a result of 

 experience only in so far as it applies to more than three 

 particles by reference to one and the same system. Hence 

 just as the dynamical time-scale is defined as a time-scale by 

 reference to which a particle free from the action of force 

 moves with a uniform speed, so the dynamical reference 

 system may be defined as a system by reference to which 

 three particles free from the action of force move in rectilinear 

 paths. Following out these considerations he finally proposes 

 to enunciate the first law in the following form : — Relatively 

 to any system of coordinates by reference to which three 

 particles projected from the same point in space and thereafter 



by reference to which the first and second laws hold, it may be proved 

 by means of the second and third laws that relatively to these axes 

 the centre of mass of a system of particles will have no acceleration, 

 provided no external forces act on the system. While, therefore, the 

 assumption that the centre of mass of the universe may be employed as 

 the origin of a dynamical reference system is justified, it is obvious 

 (1) that if, in employing Mach's expression of the first law, we restrict 

 ourselves to a part of the universe, it must be a part on which no external 

 forces act ; and (2) that since, in obtaining this form of the law and the 

 corresponding form of the second law, we employ the third law of motion, 

 the new laws are not merely new expressions of the old laws, but involve 

 the third law in addition. 



* See also Prof. Tait's solution of this problem by Quaternions in the 

 paper cited above. 



T 'Treatise on Natural Philosophy,' vol. i. part 1 (1879), § 249. 



