388.] 



CONSTRAINED SYSTEM. 



217 



displacements are arbitrary, their coefficients must also vanish 

 separately. Thus it follows that the coefficients of all the 

 displacements in the resulting equation must vanish, and we 

 have n sets of 3 equations of the type 



my = 



(4) 



r * i * 



It is apparent from these equations that the constraining 

 force acting on the particle m has the components 



387. It has thus been shown that a system of n particles 

 subject to k conditions has ^n k independent equations of 

 motion. The equations can be obtained either by eliminating 

 from d'Alembert's equation (2) k of the 3/2 displacements r, 

 Sj>, 82 by means of the k equations (3), and then equating to 

 zero the coefficients of the remaining 3 k arbitrary displace- 

 ments, or they may be regarded as represented by the 3 n equa- 

 tions (4), since these equations contain k arbitrary quantities 

 X, fj,, -. In this latter form they are sometimes denoted as 

 Lagrange s first form of the equations of motion. 



388. It follows from the remark at the end of Art. 385, that 

 the actual displacements dx, dy, dz of the particles can be 

 selected as virtual displacements only, and always, when the 

 conditional equations (i) do not contain the time. If this con- 

 dition be fulfilled, d'Alembert's equation (2) can be written 



?w(xdr+ydy+~sidz) = l(Xdx+ Ydy + Zdz), 

 or d$kmiP = 'S t (Xdx+ Ydy + Zdz). (6) 



This relation can also be deduced from the equations (4) 

 by multiplying them by xdt, ydt t zdt, and summing the equa- 



