Motion of a System of Bodies. 289 



xd^y — yd^x 

 spectively, then add the products, and we have j- =Qz— 



Py, also -17-2 =Q!z' - P'y% and so on for all the bodies m, 



m', &;c. Let a unit of m act on a unit of m' with any force p, then 

 (by the well known law of equal action and reaction ;) a unit of m/ 

 will react on a unit of m, with the force —p, which is directly oppo- 

 site top; hence mp= the whole force with which m acts on a unit 

 of w', and —m^p— the whole force of the consequent reaction of m' 

 on a unit of wz; let/ denote the straight hne which joins m and m\ 

 then evidently the forces mp, —m'p are exerted along the line/; 

 hence by resolving mp in the directions of the axes of a; and y, we 



[x—z'\ {y~y'\ 



have \'~F~ I Xmp and I ~f~ I Xmp respectively for the parts of 



— 7^ 1 X m'py — 



(y~y'\ 

 — 7=— I X m'p, are the parts of P and Q which arise from the force 



— m'p; hence, (for simplicity,) considering these forces only, we 

 have (Qx -Pt/) = ^ [—^ ]y~\ -j- j xj -mfp^ ( ^-j — j m'p, also 



QV-Py= - y~/^ ]mp, .'. we have m{Q.x-Fy)+m'{Q'x'~ 



Py)=0: it is hence evident that if we multiply the equations 



zd^y— yd^x x'd'^y' — y'd^z' 



— ^/— =Q^-Py. ^; =QV-Py, he. by m, m', 



fxd'y-yd^x\ 

 cue. respectively, and add the products, we shall have Sm ( ^r^ 1 



= Sm(Qx — Py) which is independent of the reciprocal actions of the 

 bodies on each other, for the mutual actions of every two of them 

 will destroy each other as above ; the equation which we have ob- 

 tained is the same as the first of (18), and the second and third of 

 (18) are easily found by a similar process. 



(To be continued.) 

 Vol. XXV.— No. 2. 37 



