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 x, y, and then resolved in a direction at right an- 



& r'^dv' 

 gles to r; let r', v', q= qj. ? T', denote the corresponding quan- 

 tities for m', and so on for m'', Sic. Then as at p. 134, cdc—^r^dv, 

 or, (since =dt,) dc=Trdt, in the same way dc'=Tr'dt, and 



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

 &c. (6). If the first of (6) is multiplied by to, the second by m\ 

 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 to' with the force p, then, as 

 before shown, TOp=the whole force with which m acts on a unit of 

 to', and — TO'p=^the whole force wiih which to' reacts on a unit of 

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

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

 the extremities of r and r' be joined by the straight line q, put (p, cp', 

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

 tively ; then mp' sin. 9, — m'p' sin. cp' are the forces mp', — m'p', 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 to, and 

 the second by to', then adding the products, there results the term 

 dtmm'p'{ — r sin. 9'+^' sin. 9) from the action of to and the consequent 

 reaction of to'; but the triangle, (sides r, ?•', q,) gives r \ r'W sin. 9 : 

 sin. 9', •'. — rsin. 9'+'"'sin. 9=0, which reduces the above term to 

 zero : hence mdc-\-m'dc' ■\-h,c. = dtx{mHr-\-m'T'r' -\-h,c.) 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, z and y, z; and by represent- 

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

 the plane a;, z; and by ,,0, ,,T, „r, ,,v, for the plane y, z; they 

 will be md,c-\-m'd^c' -{-hc.=dt X (to/T,^ + ^'/T'/ + &;c.) and md„c-\- 

 m'd„c'-{-hc.=dtx{m„T ,,r-{-m„T' ,/-\-hc.) Let toc+to'c'+&;c: 

 be denoted by Smc, mTr+m'T'r'-^rhc. by SmTr, and so on for the 

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

 dSm,c=dt.Sm,T ,r, dSm,^c=dt.Sm,PL^,r, (7); which are independent 

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



