58 HI. GENERAL PRINCIPLES. WORK AND ENERGY. 



Accordingly, taking account only of the terms of the first order in 

 the small quantities 8x r , 8y r , dz r , and using equations 6) we have 



If a number of particles are displaced, we must take the sum of 

 expressions like the above for all the particles, or 



as the conditions which must be satisfied by all the displacements 

 dx r , dy r , dz r . There must be one such equation for each function cp. 

 Such displacements, which are purely arbitrary, except that they 

 satisfy the equations of condition, are called virtual, being possible, 

 as opposed to the displacements that actually take place in a motion 

 of the system. If the equations of constraint contain the time, t is 

 supposed to be kept constant during the virtual displacement. 



The number of independent coordinates possessed by a system 

 is called the number of degrees of freedom of the system, which may 

 be otherwise defined as the number of data necessary to fully 

 specify its position. Between the 3n changes dx, dy, dz, occurring in 

 an equation, there are It linear equations, hence only 3^ k of them 

 may be taken arbitrarily, and this is the number of degrees of 

 freedom of the system. 



It has long been customary to make a subdivision of the subject 

 of Dynamics entitled Statics which deals with only those problems 

 in which forces produce equilibrium. A system is in equilibrium 

 when the impressed forces upon its various particles together with the 

 constraints balance each other in such a way that there is no tendency 

 toward motion of any part of the system. The Principle of Virtual 

 Work is the most general analytical statement of the conditions of 

 equilibrium of a system. It was used in a very simple form by 

 Galileo, but its generality and its utility for the solution of problems 

 in statics was first recognized by Jean Bernoulli, and it was made 

 by Lagrange the foundation of statics. 1 ) 



If the system consists of a single free particle, in order for it 

 to be in equilibrium the resultant of all the forces applied to it, 

 whose components are X = XX r , Y= I.Y r , Z = I.Z r) must vanish, 



9) x=r=z=o. 



1) For the history of the principle see Lagrange, Mecanique Analytique, 

 Partie, Section I, 16 and 17. 



