30 Composition and Resolution of Forces, &c. 
are considered as positive, then those which tend to decrease them, 
must be considered as negative. 
Again, by comparing (g) with (12) and (13), it is evident that the 
sums of the moments of the foreign forces to turn the system about 
the axes of z, y, x, must each =0, when the system is in equilibrium ; 
where it may be observed in forming these sums, that if those mo- 
ments which tend to turn the system in one direction are considered 
as positive, then those which tend to turn it in the contrary direction 
must be considered as negative. ‘The equations (f) and (g), agree 
with the equations (m) and (n), given at p. 43, of the Mecanique 
Celeste ; and if we are not greatly mistaken, they have been obtain- 
ed Sans principles altogether more simple than those used by La 
Place, or by any other author with whom we are acquainted. 
It may not be improper to observe that (g) are independent of 
any forces which act on the bodies in the directions of straight lines 
drawn to the origin of the coordinates. For let A denote the dis- 
tance of m from the origin of the coordinates, and mS any force — 
which acts on m in the direction of h, then we shall have S for the 
force which acts on a unit of m in that direction ; and by decompo- 
sing Sin the directions of the axes of x and y, we shall have 5x S, 
ix XS for the components of P and Q which depend on the force 
S$; .*. by considering these forces only, we have Py—Qr= 
(tz *) x S=0; hence we may anaes all such forces in form- 
ing (g). 
Again, since (f) and (g) have been derived from (a), (b), (c), we 
may neglect any six of the equations contained in (a), (b), (c) and 
use (f) and (g) for the neglected equations ; which with the remain- 
ing equations in (a), (b), (c), will make as many equations as there 
are coordinates, x y z, x’ y’ z’, &c. for all the bodies m, m’, &c. 
Hence we shall suppose six of the equations (a), (b), (c), to be 
neglected, and that (f) and (g) together with the remaining equa- 
tions in (a), (b), (c), are used to determine the positions of m, m’, 
&e. 
Equilibrium of a rigid system. 
_.We shall now suppose that all the bodies of the system are inva- 
riably connected together, or that it is rigid; then by considering 
