62 HI. GENERAL PRINCIPLES. WORK AND ENERGY. 



of the & other tf's vanish, then the coefficients of the 3n ~k arbitrary 

 <fs must vanish, so that we get the Sn equations 



1 i J_ 2 2 , i * A 



* + 2 " * 



Eliminating from these the ;t quantities A, we have 3^ k equations 

 expressing the conditions of equilibrium, being as many as the system 

 has degrees of freedom. 



The equations 16) were given by Lagrange 1 ), to whom the 

 principle of the use of the indeterminate multipliers I is due. 2 ) One 

 great advantage of the principle of virtual work is that it enables 

 us to dispense with the calculation of the reactions, for in a dis- 

 placement compatible with the constraints the work of the reactions 

 vanishes. 



As an example let us find the position of equilibrium of two 

 heavy particles of mass m l and m 2 , connected by a rigid bar without 

 weight, of length ?, and placed inside of a smooth sphere of radius r. 

 The equations of constraint are 



*i 2 + 2/i 2 -Mi 2 -*- 2 = 0, 

 ^ + 2/ 2 2 + ^ - r 2 = 0, 

 (x, - x^ + (y, - 2/ 2 ) 2 + & -z^-V = 0. 

 The equation of virtual work is 



m^gd^ + m^9^^ = 0, 

 where 8z and $ 2 satisfy the equations 







These are four linear equations between the six quantities 



##i, ##!> <5X, ^^, ^2/2^ ^^2- 



We may therefore take any two of them arbitrarily. Suppose we 

 assume dy l = dy 2 = 0. We then have four linear equations in 

 dx, d 17 dx 2 , d 2 , and in order that they may be satisfied for values 



1) Lagrange, Mecanique Analytique, torn. I, p. 79. 



2) See Note II. 



