2i6 MOTION OF A VARIABLE SYSTEM. [386. 



such virtual displacement. In other words, in d'Alembert's 

 equation 



o t (2) 



the constraining forces due to the conditions will not appear if 

 the displacements &r, fy, 82 be so selected as to be compatible 

 with the k conditions (i). 



Now this will be the case if these displacements be made to 

 satisfy the equations that result from differentiating the condi- 

 tions (i), viz. 



= o, (3) 



It should be noticed that in this differentiation, or rather 

 variation, the time t is regarded as constant. If, for instance, 

 one of the conditions (i) constrain a particle to a curve or sur- 

 face varying with the time, say the surface of the moving earth, 

 or that of a projectile in motion, the displacement is called 

 virtual, or compatible with the condition, if it takes place on 

 the curve or surface regarded momentarily as fixed (comp. Art. 

 193). Indeed, when the conditions contain the time, the state- 

 ment that a virtual displacement is one compatible with the 

 conditions has no definite meaning ; virtual displacements are 

 then defined as displacements satisfying the equations (3). 



386. The k equations (3) make it possible to eliminate k of 

 the 372 displacements from d'Alembert's equation (2). There 

 will remain ^n k independent arbitrary displacements, whose 

 coefficients equated to zero give the ^nk equations of motion. 



Applying the method of indeterminate multipliers (comp. 

 Art. 194) to perform this elimination in a systematic way, we 

 have to multiply the k equations (i) by indeterminate factors 

 X, /z, , and to add them to equation (2). The k multipliers X, 

 //,, can then be so selected as to make the coefficients of k of 

 the 3 displacements r, ty, z vanish. As the remaining $11 k 



