electromagnetic theory; and further, when there is no electromagnetic field our previous geometry is valid. But in the more general case we have to adopt the more general geometry in which there appear fourteen coefficients, ten describing the gravitational and four the electrical conditions of the world.
We ought now to seek the law of the electromagnetic field on the same lines as we sought for the law of gravitation, laying down the condition that it must be independent of mesh-system and gauge-system since it seeks to limit the possible kinds of world which can exist in nature. Happily this presents no difficulty, because the law expressed by Maxwell's equations, and universally adopted, fulfils the conditions. There is no need to modify it fundamentally as we modified the law of gravitation. We do, however, generalise it so that it applies when a gravitational field is present at the same time—not merely, as given by Maxwell, for flat space-time. The deflection of electromagnetic waves (light) by a gravitational field is duly contained in this generalised law.
Strictly speaking the laws of gravitation and of the electromagnetic field are not two laws but one law, as the geometry of the 's and the 's is one geometry. Although it is often convenient to separate them, they are really parts of the general condition limiting the possible kinds of metric that can occur in empty space.
It will be remembered that the four-fold arbitrariness of our mesh-system involved four identities, which were found to express the conservation of energy and momentum. In the new geometry there is a fifth arbitrariness, namely that of the selected gauge-system. This must also give rise to an identity; and it is found that the new identity expresses the law of conservation of electric charge.
A grasp of the new geometry may perhaps be assisted by a further comparison. Suppose an observer has laid down a line of a certain length and in a certain direction at a point , and he wishes to lay down an exactly similar line at a distant point . If he is in flat space there will be no difficulty; he will have to proceed by steps, a kind of triangulation, but the route chosen is of no importance. We know definitely that there is just one direction at parallel to the original direction at ; and it is
