Principle of Virtual Velocities. 79 



same orders of minuteness P+^P, Q.-f-'^Q, &c. may be supposed 

 to act ill jp, q, &c. ; hence the forces P+'^Pj Q' + ^^l, &c. in (4) 

 may be supposed to act in j9, </, &c., the distances of their points 

 of apphcation from the centres being f + ^p, q-h^fjf, &c. Again, 

 ^P, 5(i, &c. may be supposed to act at the same points and in the 

 same hnes as P, Q,, &c. in (1), .'. by changing P, Q, &c. into 

 P4-<^P, Q. + ^Q', fcc. without changing j?, q, &c., also using M4- 

 ^M' to denote what M becomes, (1) will become (P+<5P)p4- 

 (a+'^a)^+&c.=M+<J]Vr, (5). Now the forces P, d, &c. in 

 (5), which are the same as in (1), balance each other; also the 

 forces (5P, <5Q,, &,c. in (5) will balance each other when quantities 

 of the order dP . Sp^ dQ, . Sq, &c. are rejected, which ought to be 

 done on account of the supposed minuteness of <JP, SGi, &c. Now 

 the forces in (4) and (5) are the same, and act in the same lines 

 and directions (when quantities of the orders {^p)'^ , (^q)^, ^P • ^p, 

 &c. are rejected,) the points of application of P+^P, Gl-f-^Q, 

 &.C. in (4) being at the distances p + ^p, q-{-^q, &c. from their 

 centres, and at the distances p, q, &c. in (5) from the same cen- 

 tres ; but the forces in (5) are in equilibrium, .'. they ought (by 

 what has been shown before) to be in equilibrium in (4), suppos- 

 ing the corresponding points of application of each force in (4) 

 and (5) to be firmly connected ; hence we may equate the first 

 members of (4) and (5), and indeed we ought to equate them to 

 indicate that the forces are the same ; hence we have (P + ^P). 

 {p + 3p) + {Gi + SGi).{q-^Sq)^^c.=^{P + 8P)p-{-{Q. + 8Gi)q + &c., 

 or by reduction (P+^P)(^p + (Q,+^Q.)'5^+&c. = 0; or rejecting 

 quantities of the order ^P . <5p, <5Q, . 3q^ ^c. we have 'P3p-{-Gidq-{- 

 &c. = 0, (6), which is the equation of virtual velocities as re- 

 quired. It may be remarked, that in applying (6), we must put 

 the coefficient of any independent variation which may be intro- 

 duced by the variations Sp, ^q, &c. = 0, since dp, dq^ &c. are in- 

 dependent of P, Q,, &c. 



Application. — Let Q,CP be a straight lever, whose fulcrum is 

 C ; P and Q, the weight and power, 

 which tend to descend in the par- 

 allel lines PB, (^A, which are per- 

 pendicular to the horizontal plane 

 AB ; and suppose that P and Q. bal- 

 ance each other, to determine the 

 relation of P and Q., supposing CP 

 and 0,0 to be given ? 



