78 Principle of Virtual Velocities. 
positive) by M, we get Pp +Qq+Rr+&c.=M, (1). Since the 
forces balance each other, it is manifest that their action will not 
alter any of the distances p, 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, g, &c. by p’, q’, &e. 
(the centres of the several forces remaining unchanged, ) and the 
new value of M by M’, (1) will be changed to Pp’/+Qq/+&c.= 
M’, (2). It is also manifest that there are an indefinite number 
of ways in which Pp’, 7, &c. may be taken (without affecting 
any one of the forces P, Q, &c.) so that we shall have M’=M, 
or Pp’+Qq’ +é&c. =PpQy}ce. (3). If we use 4, (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=4p, 
q’ -qg=4, and so on; hence (3) is easily changed to P4p+ 
Q4q+&c.=0, (4), fohich ought to be satisfied independently of 
any arbitrary quantities which may be introduced by 4p, 4q, &e.; 
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; % &e., p’, q’, Kc. ; .”. they do not 
affect Jp, dq, &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 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, p, q, &c., P, Q, &c. become prop, ¢+5q, &e., P+9 
Q+5Q, &e.; ; also that M becomes M-+°M, then (1) will be 
changed to (POP) . (p+9p)+(Q-+5Q): (q+5q)4+&o. = M+ 
OM, (4). Sok onertge ple q+0q, &e. are equal to theit as 
4) »q, &e. (in in(1),) when quant fats ta ol 3 
(8q)°, (8p)*, &e. are rejected, also by rejecting quantities of the 
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