328 Yll;n AL VELOCITIES. EOD. 



i'.' !. when a system of forces acting on a ].u( id- 



is in equilibrium, each force is equal and opposite t<> ti, 

 Bultant of all the other forces, and, as we have just sem, the 

 sum of tin- products of each force into its virtual velo< 

 zero, it follows, th:tt the. product of any force into its \ irtual 

 velocity is numerically equal to the sura of such prod in: 

 any system of forces which it balances, but is of the opposite 

 sign. Hence if a single force is the resultant of a syst 

 forces acting at a point the product of the MirJ>- i'.uvr into 

 its virtual velocity is equal to the sum of such products for tin- 

 system of forces. 



252. Next, suppose a rigid rod acted on by a fore 

 end. Let x, y, z be the co-ordinates of one end, and x . 

 those of the other ; / the length of the rod ; th 



Suppose the rod displaced; let Bx, By, 6- be tin- 

 made in the co-ordinates of one end; Bx, By', Bz those made 

 in the co-ordinates of the other end ; then 



( x +Bx-x'-Bx'y+ ( y +fy-y-s/)'+ (z+&z- z '-Mr=r. ... (2). 



From (1) and (2), 



2 (x - x) (Bx - Bx') + 2 (y - y') (By - By') + 2 (z-z'} (Bz - Bz 



(Bz-Bz'Y=() ......... (3). 



Let a, /8, 7 be the angles which the original direction of the 

 rod makes with the axes ; then 



x x = I cos a, y - y = I cos , z z = I cos 7 ... (4). 



hen, in (3), we neglect the terms (&e-8s') f , (By -By)*, 

 (Bz Bz')* in comparison with those we retai ive 



(x-x') (Bx-Bx') + (y-y')(By-By') + (z - z') (Bz - Bz') = 0, 

 or, by means of (4), 

 Bx cos a + By cos + Bz cosy- Bx cos a+By' cos+&;'cos 7 . . . (5) . 



Suppose P the resultant of the forces .it one end of 



the rod, and P' the resultant of those acting at the "tin -r i nd; 

 then, in order that there may be equilibrium, th- 



