222 CELESTIAL MECHANICS. I. V. 22. 



§ 22. The principles of the preservation of impetus 

 and of areas will still hold good, if the origin of the 

 coordinates he supposed to have a uniform rectilinear 

 motion. In this case, the plane passing through this 

 point J on which the sum of the areas described is a maxi- 

 mum, remains always parallel to itself These properties 

 may he referred to the relations of the coordinates of the 

 mutual distances of the hodies of the system. The planes 

 which pass throuqh the centre of gravity of each hody of 

 the system, parallel to the general mean plane of revolu- 

 tion, are possessed of similar properties. P. 61 . 



331. Theorem. The constancy of the 

 impetus and of the areas described is ob- 

 served by a system of bodies, referred to a 

 common origin, which moves uniformly in a 

 right Hne. 



If we call the coordinates of the moveable origin of the 

 ordinates of the system X, Y, and Z, and suppose 



x—X + x/, y—Y+y, ; zzzZ-{-z^ 



x'—X + x'; yz=.Y-\-y\\ zf:=iZ+z'/, the coordinates 

 of m, m', . . . , referred to the moveable origin, will be 

 ^/» y,> ^/> ^',y • • • > ^^^ since by the properties of the centre 

 of gravity (322), we have for any detached system of 



(dd^ \ 



—- ;;- — P j, . . . , we obtain, by substitution 



0=Xm (d^X + d^x)—lmPdt^ ; 



0=lm {d^Y^di'~y)—XmQdt^; 



0=lm{d'^Z-\-d^z)~XmRdt^; consequently Xmd^X;— 

 XmPdt^—0, since d^XrrO by the supposition, and in the 

 same manner, Xmd^y^ — SmQdf^mO, and Xmd^z^ — XmRdt* 



=0. Then in the equation (P), 0:=iXm^x(^-^ - P ). . ., 



