Principle of Virtual Velocities. 79 
same orders of minuteness P+-6P, Q+45Q, &c. may be supposed 
to act in p, g, &c.; hence the forces P+0P, Q+5Q,, &e. in (4) 
may be supposed to act in p, q, &c., the distances of their points 
of application from the centres being p+5p, q+45q, &c. Again, 
oP, 6Q, &c. may be supposed to act at the same points and in the 
same lines as P, Q, &c. in (1), .*. by changing P, Q, &c. into 
P+6P, Q+5Q, &c. without changing p, q, &e., also using M+ 
OM’ to denote what M becomes, (1) will evan (P+5P)p+ 
(Q+9Q )g+&c.=M+0M’, (5). Now the forces P, Q, Sc. in 
(5), which are the same as in (1), balance each other; also the 
forces dP, 9Q, &c. in (5) will balance each other when quantities 
of the order 5P.. 5p, SQ . dg, &c. are rejected, which ought to be 
Ti on account of the supposed minuteness of #P, 9Q, &c. Now 
* the forces in (4) and (5) are the same, and act in the same lines 
and directions (when quantities of the orders (5p)", (9q)?, 5P.. op, 
&c. are rejected,) the points of application of PLoP, Q+46Q, 
&c. in (4) being at the distances p+dp, q+4q, &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+0P). 
(p+op)+(Q+9Q). (q¢+5q)+&e.= =(P+0P)p-+(@+0Q)g+&e, 
or by reduction y ARE AE ase Saag or rejecting 
PMH, of the order 5P . dp, 6Q .59, &c. we have Pip +Qdq+ 
&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 1 may be intro- 
duced by the variations 4p, 5g, &¢.=0, since dp, 5g, &c. are in- 
dependent of P, Q, &c. 
_ Application ict QCP be a straight lever, whose fulcrum is 
O; P and Q the weight and power, 
which tend to descend in the par- 
allel lines PB, QA, which are per- 
pendicular to the horizontal plane 
AB ; and suppose that P and @ bal- 
ance each other, to determine the @ 
and QC to be given? ee 
, 
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