68 Demonstration of the Principle of Virtual Velocities. 
applied move with the force P’, and that to which the force Q is 
applied 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’, Q’, &c. are applied in directions 
which are directly opposite to their several directions they will 
balance the forces P, Q, &c.; hence if 5p, 5g, &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, mg’ 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 
Pdp + Qdq4+ &c.4+P/mp’+Q’mq'+&c. = 0, but by supposition 
Pép+Qiq+&e.=9, - *, Pomp’ +Q/mq + &c.=0, or P’p’+Q/q/+ 
&c.=0; now it is evident that P’ has the same sign as p’, Q’ the 
game sign as gq’, and so on ; hence the equation cannot hold good, 
(since its terms have all the same sign which is +,) unless P’p’ 
=0, Q’q’=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 # 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 « 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 6 reckoned from the origin in the axes 
of x and y, such that-a is greater than 7, 6 greater than y, then we 
shall have p=a=«, g=b —y; also let.a’, b, be the co-ordinates of 
any fixed point in th tubtinaat ditnctiones 2, then evidently a’ is 
less than x, and b’ ie Fenn thetaeyés r=a/(r—a’)? +(y—0/)3; the 
