68 Demonsii'ation of ike Principle of Virtual Velocities. 



applied move with the force P', and that to which the force Q. is 

 apphed move with the force Q,', and so on, and suppose the bodies 

 or points describe the very small spaces p\ q', &c. in the same 

 time ; then if the forces P', Gi', &c. are applied in directions 

 which are directly opposite to their several directions they will 

 balance the forces P, Q., &c. ; hence if <Jjo, Sq^ &c. are the virtual 

 velocities of the points of application of P, Q, &c. if we assume 

 mp' for the virtual velocity of P^ when it is applied in a direction 

 exactly opposite to its direction, mq' will be the virtual velocity 

 of Q.' when it is changed to the opposite direction, and so on. 



Hence by (4), since the system is in equilibrium, we shall have 

 P(J/?+Q,'55'-f &c.+P'my+Q.'m5'-|-&c. =0, but by supposition 

 Pdp-i-adq+6cc.^0, .•.P'mp^+a'w9' + &c. = 0, or Py + aY+ 

 &c. = ; now it is evident that P' has the same sign as p', CI' the 

 same sign as q', and so on ; hence the equation cannot hold good, 

 (since its terms have all the same sign which is +;) unless P'p^ 

 = 0, GlY=0, and so on; .•.P' = 0, or p' = 0, or both — 0, but on 

 either supposition, the body to which the force P is applied is at 

 rest, and in the same way the body to which Q, is applied is at 

 rest, and so on ; .'. when the equation of virtual velocities has 

 place, the forces balance each other, as we proposed to prove. 



Application. 



Let P, Q,, R, be three forces applied to a material point, and 

 (for simplicity) suppose the directions of P and Q, to be perpen- 

 dicular to each other and parallel to two rectangular axes a; and 

 y, drawn in their plane through any given point taken for their 

 origin, and suppose that P and Q,, act in the directions of x and 

 y, positive ; then when there is an equilibrium between P, Q,, R, 

 it is evident that R must act in the same plane with P and Q,, in 

 a direction which is directly opposite to their resultant; also that 

 R will be of the same magnitude as the resultant. 



Let X and y be the co-ordinates of the point of application, 

 (which is supposed to be within the angle formed by the positive 

 co-ordinates,) of the forces when referred to the aforesaid axes ; 

 take the distances a and b reckoned from the origin in the axes 

 of X and y, such that a is greater than x, b greater than y, then we 

 shall have j9=a - x, q — b—y] also let a', b', be the co-ordinates of 

 any fixed point in the line of direction of R, then evidently a' is 

 less than -r, and 6' is less than y ; .'.r = \/{x—a'y'\-{y — b'Y', the 



