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Demonstration of the Principle of Virtual Velocities. 67 
portions among themselves that P, Q, &c. have ; also if any one of 
them as P’ is=0, then each of the others and M’ will =0, as evi- 
dently ought to be the case, for when a system of forces as P, Q, R, 
&c. is in equilibrium, the equilibrium will not be disturbed by ap- 
plying another system of forces, as P’, Q’, &c. which are propor- 
tional to P, Q, R, &c., to the same points severally, and in the 
same directions or in directions which are exactly opposite, &c. 
We shall use 4, (the characteristic of variations,) when prefixed 
to any quantity to denote any indefinitely small variation of the 
quantity, the variation being supposed to be positive when the 
quantity is increased, and negative when it is decreased. Sup- 
pose then that the forces balance each other, and that the body 
or system of bodies, receives a very small change of position, 
(consistent with its conditions, or with the mutual connections 
of its parts in the case of a system,) and that in consequence of 
the change of position p, g, &c. become p+4p, g+4q, and so on, 
and that P, Q, é&c. become P+45P, Q+5Q, &Kc., also that M be- 
comes M-+0M; then (1) will be changed to (P+9P).(p+¢p)+ 
(Q+5Q) . (¢+0q)+&c.=M+95M; now since SP, op, &c. are 
each supposed to be indefinitely small, the products dP . op, 
dQ . 9g, &c. will be indefinitely smaller than poP, Pp, and so on, 
and are hence to be rejected; .°. rejecting these products and re- 
ducing by (1), the above equation will become pdP + qIQ +&c. 
+ Pip +Qdq+&c. =0M, and if we assume poP +qIQ +&c.=0M, 
(3), we get Pip +Qdq+&c.=0, (4). Now it is evident (as in 
(2),) since p, g, &c. are the same in (3) as in (1), that we may 
suppose the forces 5P, dQ, &c. to be applied at the same points 
and to act in the same lines as P, Q, &c. severally, by neglecting 
quantities of the order of the products 9P. op, dQ. dq, &c.; 
hence OP, 9Q, &c. will have the same sign, and the same propor- 
tions among themselves that P, Q@, &c. have; .". when the forces 
balance each other, changing the position of the body or system 
(as above, in consequence of which, the small forces, 8P, dQ, &c. 
are introduced), does not affect the equilibrium; and (4) which 
is called the principle of virtual velocities, will have place when 
the forces P, Q, &c. balance each other, as we proposed to prove; 
and it may be observed that dp, dg, &c. are called the virtual 
mnectinie of the points of application of P, Q, &c. 
Conversely if (4) has som the forces will balance each abe 
disedtecaecactiacadh Bidirweny do the body or point to which P is 
