Let () be any such vector. Then and and are each of them functions of () and also of the time , i.e. they are functions of . We shall assume that our physical quantities are differentiable, except possibly at exceptional points. Let ) stand for (), and analogously and ) for the vector Finally if () be another vector at the same point, then stands for what is called the 'vector product' of the two vectors, namely the vector It is evident that ) can be expressed in the symbolic form The vector equation is an abbreviation of the three equations Let ) be the electric force at ), and let ) be the magnetic force at the same point and time. Also let be the volume density of the electric charge and ) its velocity; and let ) be the ponderomotive force: all equally at ). Finally let be the velocity of light in vacuo. Then Lorentz's form of Maxwell's equation is It will be noted that each of the vector equations (3), (4), (5) stands for three ordinary equations, so that there are eleven equations in the five formulae. CHAPTER III SCIENTIFIC RELATIVITY 7. Consentient Sets. 7.1 A traveller in a railway carriage sees a fixed point of the carriage. The wayside stationmaster knows that the traveller has been in fact observing a track of points reaching from London to Manchester. The stationmaster notes his station as fixed in the earth. A being in the sun conceives the station as exhibiting a track in space round the sun, and the railway carriage as marking out yet another track. Thus if space be nothing but relations between material bodies, points as simple entities disappear. For a point according to one type of observation is a track of points according to another type. Galileo and the Inquisition are only in error in the single affirmation in which they both agreed, namely that absolute position is a physical fact—the sun for Galileo and the earth for the Inquisition. 7.2 Thus each rigid body defines its own space, with its own points, its own lines, and its own surfaces. Two bodies may agree in their spaces; namely, what is a point for either may be a point for both. Also if a third body agrees with either, it will agree with both. The complete set of bodies, actual or hypothetical, which agree in their space-formation will be called a 'consentient' set.