But in the second place, the theory of relativity makes it appear probable that Mach was on the right road in his thought that inertia depends upon a mutual action of matter. For we shall show in the following that, according to our equations, inert masses do act upon each other in the sense of the relativity of inertia, even if only very feebly. What is to be expected along the line of Mach's thought? 1. The inertia of a body must increase when ponderable masses are piled up in its neighbourhood. 2. A body must experience an accelerating force when neighbouring masses are accelerated, and, in fact, the force must be in the same direction as the acceleration. 3. A rotating hollow body must generate inside of itself a "Coriolis field," which deflects moving bodies in the sense of the rotation, and a radial centrifugal field as well. We shall now show that these three effects, which are to be expected in accordance with Mach's ideas, are actually present according to our theory, although their magnitude is so small that confirmation of them by laboratory experiments is not to be thought of. For this purpose we shall go back to the equations of motion of a material particle (90), and carry the approximations somewhat further than was done in equation (90a). [Pg 106] First, we consider as small of the first order. The square of the velocity of masses moving under the influence of the gravitational force is of the same order, according to the energy equation. It is therefore logical to regard the velocities of the material particles we are considering, as well as the velocities of the masses which generate the field, as small, of the order . We shall now carry out the approximation in the equations that arise from the field equations (101) and the equations of motion (90) so far as to consider terms, in the second member of (90), that are linear in those velocities. Further, we shall not put and equal to each other, but, corresponding to the higher approximation, we shall put From (90) we obtain, at first, From (101) we get, to the approximation sought for, in which, in (117), and denote the space indices only. [Pg 107] On the right-hand side of (116) we can replace 1 + by 1 and by . It is easy to see, in addition, that to this degree of approximation we must put in which , and denote space indices. We therefore obtain from (116), in the usual vector notation, The equations of motion, (118), show now, in fact, that 1. The inert mass is proportional to 1 + and therefore increases when ponderable masses approach the test body. 2. There is an inductive action of accelerated masses, of the same sign, upon the test body. This is the term . [Pg 108]