118 SYSTEMS OF FORCES. [CHAP. VII. 



The identification of the resultant stress with the reaction 

 requires us to make the supposition that the bodies are rigid. 

 For it is then immaterial whether the action of the particles of 

 B upon the particles of A in lines passing indefinitely close to the 

 point of contact P are regarded as exerted upon the particle of A 

 at P or upon other particles of A. 



If we regard a rigid body as the limiting case of a deformable 

 body when all the numbers expressing deformations fall below any 

 assigned number, however small, then it is clear that there will 

 be a certain order of approximation to which it will be correct to 

 neglect the distinction between forces exerted by the particles of 

 B upon the particles of A and the forces exerted by the particles 

 of B upon the particle at P. 



In this way the conception of rigid bodies and the reactions 

 at their points of contact becomes of importance in Mechanics. 



124. Bodily forces and surface-tractions. Consider any 

 deformable system of particles continuously distributed so as to 

 form a body. In any infinitesimal volume dv, where the density 

 is p, we say there is a particle of mass pdv. 



The field of force due to the actions of the particles of other 

 bodies would produce in this particle a resultant acceleration /, 

 where / is the intensity of the field of force at the position of the 

 particle. 



The force exerted on the particle by the particles of other 

 bodies is accordingly pfdv in the direction and sense of /. 



Since this force is proportional to the mass of the particle, 

 the resultant of such forces on all the particles in a small volume 

 contains the mass within the volume as a factor. Such forces are 

 known as bodily forces. 



On the other hand the resultants, for any part of a surface 

 bounding a portion of a body, of the stresses exerted on that 

 portion by other portions of the body, or by portions of other 

 bodies, in contact with it over the part of the surface, are known 

 as surface tractions. The expressions for the resolved parts of such 

 tractions on small areas contain as factors the areas of the surfaces 

 across which they act. 



