Motions of a System of Bodies. 43 



ant of all the disturbing forces which affect a unit of m, when re- 

 duced to the plane <r, y 9 and then resolved in a direction at right an- 



c' fidv' 



** 



gles to r ; Jet r', ?;', ^= 2 , , T', denote the corresponding qu^n 



tides for m', and so on for m'' 9 &c. Then as at p. 134, cdc=Tr*dv, 



r 2 dv 

 or, (since =dt,) dc=Trdt, in the same way dc'=T'r'dt 9 and 



so on; hence the equations of motion are dc^Trdt, d&= r T'r / dt 9 

 &c. (6). If the first of (6) is multiplied by m 9 the second by m' 9 

 and so on for all the bodies, and the products be added, the result- 

 ing equation will, (as before,) be independent of any terms arising 

 from the actions of the bodies on each other. 



For let a unit of m act on a unit of m' with the force p 9 then, as 

 before shown, mp=\he whole force with which m acts on a unit of 

 m', and — m'p =- the whole force wifh which m' reacts on a unit of 

 m; and if mp'=the projection of mp on the plane x 9 y, then evi- 

 dently — m'p'=the projection of —m'p on the same plane. Let 

 the extremities of r and r 1 be joined by the straight line q 9 put p,*? 7 , 

 for the angles of the triangle, (thus formed,) opposite r 9 r / 9 respec- 

 tively ; then mp' sin. 9, — m'p' sin. 9' are the forces mp' 9 — m'p' 9 when ■ 

 resolved at right angles to r*, r, severally ; .' . — m'p' sin. 9' is a com- 

 ponent of T in the first of (6), and mp' sin. 9 is a component of 

 T' in the second ; hence multiplying the first of (6) by m 9 and 

 the second by m'. then adding the products, there results the term 

 dtmm'p'( —r sin. 9 / -f-r / sin. 9) from the action of m and the consequent 

 reaction of m'; but the triangle, (sides r, i J 9 q 9 ) gives r I r*: ;sin. 9 : 

 sin. 9', .\ — r sin. 9 / +r / sin. 9=0, which reduces the above term to 

 zero : hence mdc+m'dc'+hc.==dtx(mTr-{-m / T / r'-{-hc.) is mani- 

 festly independent of any terms which arise from the actions of the 

 bodies on each other. In a similar way, two other equations which 

 are analogous to the above, may be obtained ; by projecting the 

 bodies* and the forces on the planes x 9 z and y 9 z; and by represent- 

 ing the quantities corresponding to c, T, r, v, he. by ,c, ,T, ,r, ,v for 

 the plane x 9 z ; and by „c, 7/ T, „r, u v, for the plane y 9 z; they 

 will be md / c±m'd / c'-{-kc.=dt X (m,T / r-\ r m / ,T / ,r'-^hc.) and md„c-\- 

 m / rf // c / +&c.=^x(ff» // T // r+m // T , // r / +&c.) Let mc+m'c'+hc. 

 be denoted by Smc, mTr+m'T'r'+kc. by SwTr, and so on for the 

 other equations; then the above equations become dSmc=dt.SmTr, 

 </Sm,c=<fr.S»i/iy, dSm,,c=dt.Sm„T // r, (7) ; which are independent 

 of any terms arising from the actions of the bodies on each other, 



