105-109] D'ALEMBERT'S PRINCIPLE. 107 



In like manner we have 



^my = 2 F, and 2m = ^Z. 



Again multiplying the ^-equations by the y's and the ^-equa- 

 tions by the z's, adding, and remembering that 2 (yZ' zY') 0, 

 we have 



In like manner we have 

 2m (zx - xz) = 2 (*Z - arZ), and 2ra (y - yaj) = 2 (o?F- 



Our equations may be stated in words : 



(1) The sum of the resolved parts in any direction of the 

 kinetic reactions of a system of particles is equal to the sum of the 

 resolved parts of the external forces in the same direction. 



(2) The sum of the moments about any axis of the kinetic 

 reactions of a system of particles is equal to the sum of the 

 moments of the external forces about the same axis. 



The result may also be briefly stated in the form : When the 

 external forces are regarded as localised in their lines of action the 

 kinetic reactions and the external forces are two equivalent 

 systems of localised vectors. 



This result, in a slightly different form, was first stated by 

 D'Alembert in his Traite de Dynamique, 1743. It is known as 

 D'Alembert's Principle. 



108. Motion of the centre of inertia. Since the resultant 

 of the kinetic reactions of the particles of a system is the kinetic 

 reaction of a particle of mass equal to the mass of the system 

 placed at the centre of inertia and moving with it, we see that 



so that the centre of inertia moves like a particle of mass equal to 

 the mass of the system under the action of the resultant of all the 

 forces applied to the system. 



109. Motion about the centre of inertia. In the equa- 

 tions such as 2m (yz zy) = 2 (yZ zY) put x = x 4- x f , .... The 

 left-hand member of the equation just written becomes 



