140 THEORY OF WORK AND ENERGY. [CHAP. VIII. 



From the theorem of Article 140 it follows that the virtual work of the 

 forces between the particles of a rigid system is an infinitesimal of a higher 

 order than the virtual displacements. This work may therefore be omitted 

 in the calculation of the virtual work for the system. Hence for a system of 

 rigid bodies in equilibrium we have the result that the sum of the virtual 

 works of all the external forces applied to all the rigid bodies vanishes for 

 every possible infinitesimal displacement. 



*158. Virtual moment of localised vector. In the same way in which 

 we defined the virtual work of a force we may define a quantity connected 

 with any vector localised at a point. Such a quantity will be called the 

 virtual moment of the vector. Further, the definition may be made to apply 

 to a vector localised in a line through the point. We shall say that the 

 product of any infinitesimal displacement of a point and the resolved part, 

 in the direction of the displacement, of a vector localised in a line through 

 the point is the virtual moment of the vector for that displacement. 



The virtual moment of a force for any displacement is the virtual work of 

 the force in that displacement. 



It is clear, as in Article 131, that the virtual moments of two equivalent 

 systems of vectors localised at a set of points, or in lines through the points, 

 are equal. 



*159. Variational equation of motion. Since the kinetic reactions of 

 a system of particles and the forces acting upon them are equivalent systems 

 of vectors, it follows that, for every infinitesimal displacement, the virtual 

 moment of the kinetic reactions is equal to the virtual work of the forces. 



The virtual work of the forces is the work done in the infinitesimal 

 displacement, and this is equal to the infinitesimal change that would be 

 made in the kinetic energy if that displacement were executed. 



It follows that the infinitesimal change in the kinetic energy is equal to 

 the virtual moment of the kinetic reactions. 



Analytically, the virtual displacement 8x, 8y, 82, of any particle can be 

 expressed in terms of the quantities 6, 0,... that define the position of the 

 system and their differentials 86, 80,.... The kinetic reactions mx, my, mz, 

 can be expressed in terms of 0, 0,... their velocities #, 0,..., and their 

 accelerations 6, 0,..., and the equation stated above can be written 



2m (X8x + ?% + z8z) = 2 [(X+ X ) fa + ( Y+ Y ) by + (Z+ Z } 8z]. 

 The left-hand member can be transformed into an expression linear in 80, 

 80,... with coefficients depending on 0, 0,..., 0, 0,..., 0, 0,..., and the right- 

 hand member can be transformed into an expression of the form 08$ + &amp;lt;l&amp;gt;80 + 



To secure the equality of the two sides for all values of the ratios 86 : 80 : ..., 

 it is necessary and sufficient that the coefficients of 80, 80,... on the two 

 sides should be equal. 



The equations thus obtained must be the equations of motion of the 

 system expressed in terms of the quantities 6, 0,.- which define its position 



