99-] D'ALEMBERTS PRINCIPLE. 53 



respectively to the moments of the tangential and normal components 

 of the resultant force F. 



97. The Principle of d'Alembert. Let us consider a particle 

 of mass m moving under the action of any forces F v F%, F n , 

 whose resultant is F. The total acceleration j of the particle 

 has the components d*x/dfi, d^y/dP, d^z/dP parallel to the 

 rectangular axes Ox, Oy, Oz. If the forces F v F 2 , F n be 

 imagined removed, a force equal to mj would be required to give 

 the particle the same acceleration j that it had under the action 

 of the forces F v F 2 , F n . This fictitious force, mj, whose com- 

 ponents are md^x/dt^, md^y/dfi, md^z/dfi, is called the effective 

 force. For the sake of distinction, the forces F v F 2 , > F n , which 

 actually produce the motion, are called the impressed forces 

 (comp. Art. 36). 



98. The ordinary equations of motion of a particle, 



i r d^z ~ / ^ 



Y ' m W* =Z ' (26) 



where X, Y, Z are the components of the resultant F of the 

 impressed forces, express merely the equality between the 

 effective force mj and the resultant impressed force F. It fol- 

 lows that, if the reversed effective force mj, or its components, 

 m&xfdP) md^y/dfi, md^z/dfi, be combined with the 

 impressed forces F p F 2 , F n , we have a system in equilibrium. 

 This is the fundamental idea of d'Alembert's principle. 



99. The reversed effective force, mj, is sometimes called the 

 force of inertia of the particle. To understand the idea underlying 

 this expression, imagine the impressed forces to be removed, and then 

 push the particle with the hand so as to give it the same motion that it 

 had under the action of the impressed forces. The pressure of the 

 hand on the particle must at every instant be equal to the resultant F, 

 or to the effective force mj, while the equal and opposite pressure of 

 the particle on the hand represents the force of inertia. It must, how- 

 ever, be clearly understood that this force of inertia, or inertia- resist- 

 ance, is a force exerted on the hand and not on the particle. 



