1^12 



CELESTIAL MECHANICS. I. V. 21 



/_Kv 



?/) p(y'—y) 



■\- ^ y\ and mQx-Vm'Qa; 



x' ; but these two sums being equal, their difference 2m 

 (Py—Qx) vanishes:] and the same is, therefore, true re- 

 specting all the other reciprocal actions of the system, and 

 with respect to all these the sum ^m {Py — Qx) vanishes. 

 Again, if *S^ be the force which urges m towards the origin 

 of the coordinates, we shall have, as far as this force alone 



— Sx 



is concerned, F — 

 -Sy 



-, and Qz=. 



>s/ {xx -\-yy -\- zz) 



consequently Py=i Qx, and their diffe- 



cz=:l.m 



e'-^m 



s/{xx^-yy^zz) 

 rence vanishes. When., therefore, the bodies are only sub- 

 jected to their mutual action, and to the forces directed 

 to the origin of the coordinates, we have 



xAy — ydx , xdz — zdx 



d^ ;czz2:m— ^^— ; 



d^ • ^^^ 



[m) If we suppose the place of 

 Ay the body m to be projected on 

 the common plane of x andy, 

 the fluxion i (xdj/—- ydx) will 

 represent the area traced by 

 the radius drawn, from the ori- 

 gin of the coordinates, to the 

 projection of m : it follows, 

 therefore, that the sum of the areas described by the radii, 

 belonging to the different bodies of the system, multiplied 

 by their masses, is proportional to the fluxion of the time, 

 and, for any finite interval, proportional to the time itself. 

 This constitutes the principle of the constancy of areas, 

 which is obviously true for any plane whatever, since the 



y 



