OF THE MOTION OF A SYSTEM. 207 



supposed cause, since it shows that the phenomena may 

 be deduced from the operation of a constant force, acting 

 equally upon the moving body, whatever its direction 

 might be, and fulfilling the condition, that " fattraction 

 est comme une fonction de la distance, ainsi que nous le 

 supposerony ioujours*" P. 58.] 



§ 20. Of the principle of the preservation of the motion 

 of the centre of gravity : which is true even when the 

 bodies exert abrupt actions on each other, P. 54. 



322. Theorem. The centre of gravity of 

 any system of bodies perseveres in its state of 

 rest or uniform rectilinear motion, notwith- 

 standing any reciprocal action between the 

 bodies. 



If we substitute, for the variations of the places of all 

 the bodies m\ m" , . . ., the variations of the place of m 

 augmented by the difference of the variations, and make 



ga:"= ^x + ^x'\ hf-^ + V/ ^^"= ^^ + ^z\ 

 substituting these values in the expressions for tlie varia- 

 tions of/,/', . . . , the distances of the bodies (307); it is 

 obvious that Sx, ^y, ^z will disappear from these expres- 



. r ^. ^ r 2(x'-x)(^x'-gx) 2{X'^X)^, .^^^.-, 



sions [ ; thus c^ / = -^^ -\ -•=.— — -: — ■ tx . (307)1 



/ f 



Now if the system is at liberty, none of its parts being con- 

 nected with any foreign bodies, the conditions, relating to 

 their mutual connexion, depending only on their distances 

 from each other, the variations ^x, ^y, ^z, which relate to 

 a quiescent point, will be independent of these conditions; 

 whence it follows, that if we substitute these values of 

 the variations in the equation (P) (317), we may suppose 



