frames imply new forces, and provided that we are allowed to assume such concealed masses as we need.

I will end this chapter by trying to make clear the difference between the laws of motion and the special laws of nature, such as those of electricity or magnetism. or heat. We shall then see that, on the traditional view, the law of gravitation occupies a curious position, intermediate between the two sets of laws.

The

The laws of motion do not profess to tell us in detail how motions are caused or modified. What they do is to tell us the general conditions which all motions, however produced, must conform to. They take no account of the kind of matter which is moved, or of its physical or chemical state at the time; the one property of matter, other than purely geometrical properties, which appears in the laws of motion is inertial mass. special laws of nature, on the other hand, tell us about the various causes of motion. They have to take into account all sorts of properties of bodies beside their inertial masses. They have to consider whether they be electrically charged or not, whether they be hot or cold, magnetised or unmagnetised, and what sort of medium surrounds them. Now, the law of gravitation, on the traditional view, is in one way like a special law of nature, and, in another way, more like the general laws of motion. It professes to tell us one of the causes which start and modify motions. So far it resembles a special law of nature. But the only property of matter that it has to consider is common to all matter, viz. gravitational mass. And this proves to be identical with the one property which is considered in the laws of motion, viz. inertial mass. Thus there seems to be a very much closer connexion between the laws of motion and the law of gravitation than between any of the special laws of nature and the laws of motion. Again, if we are in earnest with the Relational Theory of Motion, we must suppose that all the motions with which Mechanics deals take place with respect to

material axes. And, since all matter attracts all other matter gravitationally, on the traditional view, all bodies will be attracted more or less by the axes to which their motions are referred. It thus seems not unlikely antecedently that there should be a very close connexion between the laws of motion and the law of gravitation, and that a completely Relational system of Mechanics should contain a theory of gravitation. The details of this are reserved for the next chapter, but it is hoped that the foregoing discussion of the traditional laws of motion and gravitation may have brought the reader into a proper frame of mind for understanding and criticising the General Theory of Relativity.

The following additional works may be consulted with advantage:

B. A. W. RUSSELL, Principles of Mathematics, vol. i, Part VII.
E. MACH, Science of Mechanics.

H. POINCARÉ, La Science et PHypothèse.

P. PAINLEVÉ, Les Axiomes de la Mécanique. (Paris. Gauthier

Villars.)

CHAPTER VI

"What's the use of Mercator's North Poles and Equators, Tropics, Zones, and Meridian Lines?"

So the Bellman would cry; and the crew would reply; "They are merely conventional signs!"

"Other maps are such shapes, with their islands and capes! But we've got our brave Captain to thank,"

(So the crew would protest), "that he's bought us the bestA perfect and absolute blank!”

(LEWIS CARROLL, The Hunting of the Snark.)

Modification of the Traditional Kinetics (continued). (2) The General Theory of Relativity. Summary of Part I

IN the last chapter we treated the traditional laws of motion without reference to the kinematic results of the Special Theory of Relativity, outlined in Chapter IV. That is to say, we combined the traditional Kinetics with the traditional Kinematics. We must now take a step forward, and show that the traditional laws of motion are not compatible with the modified kinematics. of even the Special Theory of Relativity. We shall then be able to advance to the General Theory.

There is no need for me to treat the kinetics of the Special Theory in any detail, because it is only a half-way house to the General Theory. I will therefore content myself with a single example to show that the traditional laws of motion cannot be reconciled, without modification, with the kinematics of the Special Theory and with the Restricted Physical Principle of Relativity.

Let us suppose that two sets of observers were doing

experiments to determine inertial mass by the impact of bodies, as described in the last chapter. One shall be on the platform p, and the other on the platform, of Chapter IV. These platforms are in uniform rectilinear relative motion in a Newtonian frame. The velocity of the first with respect to the second, as measured by observers on the second, is v12. Let two bodies be moving along 1 in the direction in which p1 is itself moving relatively to P. Let their velocities relative to 1, as measured by observers on it, be U, and u1 respectively, before they hit each other. After they

have hit, let their velocities with respect to p1 be W1 and w1 respectively. Let the observers on 1 ascribe to these bodies the inertial masses M, and m, respectively. As we saw in the last chapter,

Each body has its own coefficient, which it keeps when its velocity is altered by the collision, and which is independent of its initial velocity. There is no doubt that this is very approximately true under ordinary conditions of experiment; the question is whether it can be exactly true, consistently with the Physical Principle of Relativity and the kinematics of the Special Theory.

Let the whole experiment be also watched by the observers on P. Let the velocities which they ascribe to the bodies relatively to P2 be U2, 2, W, and w respectively. The Physical Principle of Relativity tells us that if equation (1) expresses a genuine law of nature in terms of the observations of people on P1, the people on, must be able to find an equation of precisely the same form in terms of their observations on the same phenomena. That is, they ought to find that their observed relative velocities are connected by an equation M2U2+m ̧μ2 = M2W2+m2W2•

2 2

2

(2)

In this equation M, and m, will have to be independent of the velocities of the bodies; for it is obvious that

the form of the law would not be the same for both sets of observers, if, in the one case, the coefficients were constants, and, in the other, were functions of the velocity of the body.

Now it is easy to see that anything of the kind is inconsistent with the kinematics of the Special Theory of Relativity. If the reader will look back to equation (3) in Chapter IV he will see that

with similar equations, mutatis mutandis, for u,, W, and W2. It is quite obvious that, if these values be substituted in equation (2), we shall reach a result which is inconsistent with equation (1), on the assumption that the masses are independent of the velocities. It follows that the traditional view that mass is independent of velocity cannot be reconciled with the Physical Principle that genuine laws of nature have the same form for all observers who are in uniform rectilinear relative motion, and with the kinematics of the Special Theory of Relativity. It is not difficult to see what modification is needed. Let us denote by M1u the mass which has to be assigned to a body moving with a measured velocity U1 relatively to the Newtonian frame p1. Let us put

1. U

where M, and m。 are independent of the velocity. Let us then see whether the equation