I94-] VARIABLE CONSTRAINTS. IO 5 



Some authors confine the term virtual displacement to dis 

 placements compatible with the given conditions. 



The displacement &$= ^/8x 2 + fy/ 2 + Sz 2 will be compatible 

 with the conditions (i) at the given time t if the following 

 conditions, obtained by differentiating the equations (i), are 

 satisfied : 



$>x -f- fatyr + <f) g z = o, 

 ^ a &tr + ^fiy + ^fe = 0. ^ 



It should be noticed that in this differentiation the time t is 

 regarded as constant, the displacement being taken to occur at 

 a given time. 



The equations (2) and (3), which must be fulfilled simulta- 

 neously, constitute the equations of motion of our problem. 



By means of the equations (3), two of the component dis- 

 placements &r, Sy, Sz can be eliminated from the equation (2) ; 

 the coefficient of the third equated to zero gives the actual 

 equation of motion. 



194, To perform this elimination systematically the method of 

 indeterminate multipliers can be used as follows. Multiplying 

 the equations (3) respectively by the indeterminate factors X 

 and p and adding them to the equation (2), we obtain the single 

 equation 



in which the arbitrary quantities X, p can be so selected as to 

 make the coefficients of two of the three displacements 8x, fry, 

 vanish ; the coefficient of the third must then also vanish, 

 The equation is therefore equivalent to the following three 

 equations : 



(4) 

 viz = Z 



