Lagrange’s proof of virtual velocities.

51

shew conversely, that it is possible to move W, which (as wehave proved) cannot be done, however much we alter W inmagnitude.

Hence, if cip, cp 2 . be the spaces through which

«! «•;>.. move in consequence of the indefinitely small dis-

placement, those being reckoned positive when the blocks ap-proach, or string is given off, and the others negative. Thenn x 8p x , n 2 $p 2 , will be the lengths of string given off or taken onthe wheels, according as they are positive or negative ;

W]cSpi + n 2 lp 2 + . = 0,

or P 1 8p 1 + P 2 $p 2 + .= 0,

which is the Principle of Virtual Velocities.

82. The displacements 8p l , Sp 2 .must be taken

indefinitely small, otherwise the equilibrium will be sensiblydisturbed, and W will not remain at rest. In fact thebest way of representing the principle is this; that when anypart of the system is moved through a space less than anyassignable quantity, then W will move through a small spacewhich varies as the square or some higher power of the dis-turbance, so that it vanishes in the limit.

Prop. To obtain the equations of equilibrium of a rigidbody from the Principle of Virtual Velocities.

83. By this principle we have 2. Pep = 0. Let XYZbe the resolved parts of P : and Sx, Sy, Sz the virtual velocitiesof the point (xyz) with respect to P ;

.-. 2.(AT&» + Y8y + Z8x) = 0.

Now, by Art. 73, we must put

8x = a + yO — %<p, 8y = b + z\js — x9, 8% = c + , v<p — y\]s,

in which a, b, c, 9, (p, f are arbitrary small quantities: hence«2.X + 62. r+cS.Z

+ \//2. (Yss - Zy) + <j£>2. (Za? - Xz) + 82. (Xy - Vw) - 0,and because a, 6, c, 9, <p, 'js are arbitrary,

2. = 0, 2. F = 0, 2. Z = 0,

2. (F* - Zy) =0, 2 .(Zoo- Xz) = 0, 2. (. Xy - Yx) = 0,

which are the six equations of equilibrium deduced in Art, 65.