208 CELESTIAL MECHANICS. I. V. 20. 



either ^x, Sy, or 8z to subsist alone, so that its coefficients 

 will vanish: we have thus the three equations OziSm 



(^-P).0=X.@-Q),0=X.(f-«). Nowsup. 



posing X, Y, and Z to be the three coordinates of the centre 



7 .77} 'X* 7 fTl^l 



of o^ravity of the system, we have X=---^ :yi= — ^; Z— 



Xmz ^, , 1 1 XT Smddx , ^ ddX 

 : consequently, since ddX=:--- , we haveOzi 



—ImP ^ ddF XmQ , ^ ddZ XmR 



"T — ' "-:t7^ ^^ — ' ^^'^ ^-17^ ?^> ^^ ^^^^ 



z,m at- Hm qV Lm 



the motion of the centre of gravity of the system is the 



same, as if all the bodies, and all the forces acting on them, 



were united in it. (264). 



If the system is only subjected to the mutual actions of 

 the bodies composing it, we shall have 



0=zXmP', OzzSmQ; OznHmR ; 

 For if we express the mutual action of m and m' by p, and 

 their distance by /. we shall have, as far as this action 

 alone is concerned, 



^p^^r ^Q^tol). ruR^p^y 



^p,jpi±~f), ^g^PM^\^ m^R^P^, 



Hence mP + m'P'^O; mQ + m'Q'izO; mR^-m'R'-O: 

 the mutual actions of the bodies in the respective directions 

 obviously destroying each other: and it is manifest that 

 these equations would be equally true ifj9 represented any 

 finite and instantaneous action. We have also, in the ab- 

 sence of any foreign force, 



0=-^— ,0=— — , 0=.^; and by taking the fluent 



twice, X=a + 6^ Yzza' ■\-h't, and Z=a"-|-6''#, the as and 



