Composition and Resolution of Forces, &c. 89 
=) m’p, P’y’ -Qr'=— (SS ) mp, »*.m (Py—Qr) + 
m’ (P/y’ — Q’x’) =0, (e). 
Multiplying the first of (a) by m, the second by m’, and so on for 
all the bodies, then adding the products we have mP+-m/P’+-, &c. 
=0, or denoting mP+m’P’+, &c. (for brevity,) by SmP, we have 
SmP=0, which is independent of the actions of the bodies on each 
other, for the actions of every two of them will destroy each other 
as in (d); in the same way by (b) and (c), we have SnQ=0, SmR 
=0, which are independent of the actions of the bodies on each oth- 
er, as before. 
Again, multiplying the first of (a) by y, and that of (b) by —2, 
then adding the products we have Py—Quz=0, in the same way by 
the second of (a) and (b), we have P’y/— Q‘a/=0, and so on for 
every corresponding two of (a) and (b); multiplying the first of 
these equations by m, the second by m’, and so on for all the equa- 
tions, then adding, the products, and (for brevity,) denoting m 
(Py —Qr) +m'(P’y’ —Q’z’)+ &e. by Sm(Py—Qrzr), we shall 
have Sm(Py —Qr)=0, which is independent of the actions of the 
bodies on each other, for the actions of every two of them destroy 
each other, as in (e); in the same way by (a) and (c), we have 
Sm(Pz— Rzx)=0, also by (b) and (c) Sm(Ry— Qz)=0, which are 
independent of the actions of the bodies on each other, as before. 
Hence collecting the results, we have SmP=0, SmQ=0, SuR=O, 
(f); Sm(Py —Qr)=0, Sm(Pz—Rxr)=0, Sm(Ry—Qz)=0, (g); 
since these equations are independent of any actions of the bodies on 
each other, we shall suppose these forces to be neglected in forming: 
the values of P, Q, R, P’, &c. which they involve. Again, if any 
body of the system is to be in equilibrium on any surface or line, by 
subjecting the body to these conditions, the resultant of all the forces 
which press the body against the surface or line, will be destroyed 
by the equal and contrary reaction; .". we may neglect all such for- 
ces in forming the equations of equilibrium. Hence neglecting the 
forces which depend on the particular conditions of the system, and 
the actions of the bodies on each other, it is evident by (f) that the 
sums of the remaining forces (or forces foreign to the system,) de+ 
composed in the directions of the axes of x, y, z must each=0, when 
the system is in equilibrium ; where it may be observed in forming 
these sums, that if the forces which tend to increase the coordinates 
Vor. nine —No. |. 12 
