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, Gl, R, ife'C, to the same points severally, and in the 

 same directions or in directions which are exactly opposite, &c. 



We shall use ^, (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, q, &c. become p-f^jt?, q+8q^ and so on, 

 and that P, Q., &c. become P+<^P, U + '^Q, &c., also that M be- 

 comes M+t>M; then (1) will be changed to (P + <5P). [p^dp)^ 

 (a+')^a). (^+'^g)+&c.^M+<5M; now since 8V, Sp, &c. are 

 each supposed to be indefinitely small, the products dP . Sp^ 

 60, . Sq^ &c. will be indefinitely smaller than pSP, P<Jp, and so on, 

 and are hence to be rejected ; .'. rejecting these products and re- 

 ducing by (1), the above equation will become jp^P-(-5'<yQ,4-&c. 

 ■i-FSp-{-GiSq + ^c,=.8M, and if we assume p(JP+^(Ja-|-&c. = «5M, 

 (3), we get 'PSp-\-Q,Sq-{.^c. = 0, (4). Now it is evident (as in 

 (2),) since p, q, (fee. are the same in (3) as in (1), that we may 

 suppose the forces <^P, 5Q., &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 ^P . Sp^ dQ, . Sq, &c. ; 

 hence ^P, <5Q., &c. will have the same sign, and the same propor- 

 tions among themselves that P, d, &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, <5P, ^Q,, &;c. 

 are introduced), does not afiect 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 Sp^ 8q^ &,c. are called the virtual 

 velocities of the points of application of P, Q,, &c. 



Conversely if (4) has place, the forces will balance each other. 

 For if they do not balance, let the body or point to which P is 



