72-77] INTERACTION. 89 



Let / be the magnitude of the acceleration contributed to a 

 particle of mass m by the action of a particle of mass ra', this 

 acceleration is a vector localised in the line joining the particles. 

 We define the force exerted by the particle m' upon the particle m 

 to be a vector localised at a point (the position of m), of magni- 

 tude ra/*, and having the direction and sense of the acceleration / 



The line joining the particles is called the line of action of the 

 force. 



The definition includes the statement that the force exerted by 

 m on ra' is equal and opposite to that exerted by m' on ra. 



76. Resultant Force. Since, by the definition, a force is a 

 vector localised at a point, there can be no proper resultant of a 

 system of forces except when they act on a particle. Nevertheless 

 it is convenient to regard a system of forces in general as equiva- 

 lent to other systems, in the same way as if the forces were vectors 

 localised in their lines of action. We can thus determine for any 

 system of forces a resultant force at any origin and a resultant 

 couple exactly as was done for vectors localised in lines in Article 

 27. We shall see hereafter that when a force acts on any particle 

 of a rigid body it produces the same acceleration in all points of 

 the body as it would do if acting on any other particle of the 

 body lying in the line of action of the force. The force- and 

 couple-resultants of a system of forces regarded as vectors localised 

 in lines are therefore the resultant force and couple for the same 

 system offerees acting on a rigid body (See Article 115). 



77. Motion of a body. We shall show hereafter that for 

 any body there is a certain point, known as the centre of mass or 

 centre of inertia, which moves like a particle of mass equal to the 

 mass of the body acted upon by a force, which is the force that 

 would be the resultant of the system of forces applied to the body 

 if the body were rigid. 



The centre of inertia of a homogeneous spherical body is its 

 centre of figure. All the points of a rigid body moving without 

 rotation, have the same acceleration as its centre of inertia. Thus 

 (assuming the system of definitions and rules given to be applic- 

 able to natural bodies) it is easy to devise cases in which the 

 motion of the centre of inertia of a natural body can be observed. 



