78 Principle of Virtual Velocities. 



positive) by M, we get Pp+Q,5'+E'?'+&c.=M, (1). Since the 

 forces balance each other, it is manifest that their action will not 

 alter any of the distances j9, q, r, &c. 



Again, we may evidently conceive the points of application of 

 the forces in their several lines of action to be changed, provided 

 each new point of application is considered as firmly connected 

 with the point to which the corresponding force was at first applied. 



Hence if we suppose the points of application of P, Q,, &c. to 

 be changed, and denote the new values of p, q^ &c. by p', q\ &c. 

 (the centres of the several forces remaining unchanged,) and the 

 new value of M by M', (1) will be changed to Vp' +Qiq' -\-&lc.= 

 M', (2). It is also manifest that there are an indefinite number 

 of ways in which p'^ q', &c. may be taken (without affecting 

 any one of the forces P, d, &c.) so that we shall have M'=M, 

 or P/)'-l-Q,^'+&c.=Pp + Q-5' + &'C. (3). If we use -^^ (the char- 

 acteristic of finite differences,) when prefixed to any quantity, to 

 denote any finite increment or decrement of the quantity, which 

 is to be considered as positive when the quantity is increased, 

 and negative when it is decreased, we shall have p' —p~^p, 

 qi-q — Jq^ and so on; hence (3) is easily changed to P^p-}- 

 Q.z/gr4-&,c. = 0, (4), which ought to be satisfied independently of 

 any arbitrary quantities which may be introduced by ^p, ^q, &c. ; 

 that is, the coefficient of each arbitrary independent quantity thus 

 introduced must be put =0; as is evident from the consideration 

 that P, Q,, &c. do not affect p, q, &lq,., p', q', &c. ; •". they do not 

 affect ^p^ ^q, &c. 



We shall use 8, (the characteristic of variations,) when prefixed 

 to any quantity to denote any indefinitely small variation of the 

 quantity, the variation being considered as positive when the 

 quantity is increased, and negative when it is decreased. The 

 forces being supposed to be in equilibrium, let the position of the 

 body or system be very slightly changed, (but in a manner con- 

 sistent with its conditions, or with the connections of its parts in 

 the case of a system ;) and that in consequence of the change of 

 position, j3, q, &c., P, d, &c. become p+^p, q-\-^q, &c., P+<JP, 

 a + '^a, &c. ; also that M becomes M+<^M, then (I) will be 

 changed to {P+8F) . (^+Jp) + (Gl + 'Ja) • (g+<Jg') + &c. = M + 

 8M, (4). It is manifest that j7 + ^p, q + 8q, &c. are equal to their 

 projections on p, q, &c. (in ( 1),) when quantities of the orders {Spy, 

 l^qy, ((5p)3, &c. are rejected, also by rejecting quantities of the 



