88 MASS AND FORCE. [CHAP. V. 



We shall be much occupied in what follows with the theory of 

 the motions of systems of such particles ; but, keeping at present 

 to the conception of a body as made up of particles, we regard 

 bodies as bounded by surfaces and continuously occupying finite 

 volumes, and we say that at any point within the surface of the 

 body there is a particle of the body. 



73. Density. Any part of a body (so conceived) will have a 

 certain mass, and it may be part of our conception that whatever 

 part we take, the mass of the part has a constant ratio to the 

 volume. When this is so we say that the body is homogeneous, 

 and call the ratio in question the density of the body. When 

 the ratio .of the mass of a part of the body to its volume is 

 conceived as variable from part to part of the body, then, taking 

 a series of parts of diminishing volumes containing the same 

 point we get a series of numbers for the ratios of their masses 

 to their volumes; the limit of this series when the volume is 

 indefinitely diminished is defined to be the density of the body at 

 the point. It is clear that, if p is the density of a body at any 

 point, and dv an infinitesimal volume containing the point, then 

 fpdv, the integration being taken throughout the volume occupied 

 by the body, is the mass of the body. 



74. Mutual actions of Bodies. We conceive of the 

 particles of bodies as acting one upon another so as to produce 

 accelerations relative to a frame, and we conceive that the 

 acceleration of any particle relative to the frame is the resultant 

 of component accelerations contributed by the actions of other 

 particles, and then the system of component accelerations of the 

 different particles is always a system consisting of pairs of ac 

 celerations, the accelerations of a pair being the contribution / of 

 a particle (of mass ra ) to the acceleration of another particle (of 

 mass m), and the contribution / of the particle of mass m to 

 the acceleration of the particle of mass m . The accelerations/ 

 and / are localised in the line joining the particles m and m , 

 have opposite senses, and have magnitudes such that mf=m f. 

 The product mf or m f is taken to measure the mutual action of 

 the two particles. 



75. Force. We say that the particles of bodies exert forces 

 one on another. 



