The volume-density, at a time in the -space of a time-system , of the distribution of any absolutely extensive quantity mediately possessed by a material object is calculated according to the preceding definition for the case of immediately possessed quantities, except that the 'quantity mediately possessed by (or by an extensive component of ) at the time ' must be substituted everywhere for the 'quantity possessed by (or by an extensive component of ).' 59.5 We can compare the volume-densities and of an absolutely extensive quantity for two time-systems and respectively at a given event-particle , assuming, as we may assume, that the motion of the material object possessing (mediately or immediately) the quantity is regular. Let be the time-system in which the object is stationary at , and let be the volume-density at for the time-system . Let be the moments in , , and respectively which contain . Let be the measure of a small volume in which contains [and therefore the measure of the volume in the timeless -space which this instantaneous volume occupies]. Consider the event () stationary in of which this small volume is a normal cross-section, and bounded by terminal moments ′ and ″ on either side of and both near . Then, by the theory of regular motion, we can take this stationary event () as the situation of an extensive component of , when is small enough and the duration bounded by ′ and ″ is short enough. Let and be the measures of the volumes which are the oblique cross-sections of made by and . Then ultimately , , and are expressions for the measure of the quantity possessed by . But by equation (1) of 57.2 of this chapter, Now take the mutual axes for and , and let () and () be the coordinates of in and respectively, and let () and () be the velocities due to it in and respectively. Then by equation (1) of 52.6, 59.6 Now let denote differentiation following the motion () at () and let denote differentiation at the point (). Then it is easily proved that Hence from equation (2) of 59.5 above Again by using the formulae of article 52, we can prove that From these results we immediately deduce Now the condition that the total extensive quantity which is the 'charge' of any extensive component never varies when conceived as distributed through the -space is This is the well-known equation of continuity. Now equation (4) shows that if this equation holds for the space of any time-system, it holds for the spaces of all time-systems. When the equation of continuity holds, the 'charge' of any extensive component of the material object under consideration never varies. Hence it is a mere matter of words and definition whether the charge is said to be mediately possessed by the object or immediately possessed. [8]Cf. subarticle 37.3, Chapter X, Part III. [9]Cf. Principia Mathematica. CHAPTER XVI CAUSAL COMPONENTS