On the conservation of momentum

1. Scope of the paper . The present paper is a continuation of an earlier sequence in these Proceedings , entitled ‘ The inverse square law of gravitation, I, II and III’ (Milne 1936, 1937c), which itself was connected with another sequence entitled ‘Kinematics, dynamics and the scale of time, I, II and III ’(Milne 1937 a,b ). In the latter sequence the relations between t -dynamics and r -dynamics were obtained. In the former sequence a formula for the gravitational potential energy of two particles was isolated which was Lorentz-invariant for transformations from any one fundamental observer in the expanding substratum to any other fundamental observer; and the corresponding equations of motion in t -measure were obtained. In the present paper I obtain relations which correspond to the integrals of linear and angular momentum in the many-body problem of classical gravitational theory. I conclude that Einstein’s form of the conservation of linear momentum in special relativity holds good in the universe at large only in the case of a collinear set of particles moving along their line of collinearity, and I give the modification of the law of conservation which should hold good in other cases.