46 Statics. 



FG" = G'G. FB" = B'B, FA" = A'A, 

 whence 



m x A A 4- n X B'B + o X C C 



Cr Cr = - - - - 



m + /i + o 



that is (by considering the masses m, w, o, as representing the for- 

 ces, thevelocity u being common), the distance of the centre of gravity 

 of several bodies from an assumed straight line, is found by dividing 

 the sum of the moments of these bodies (taken with respect to this 

 line) by the sum of the masses. 



Let us now conceive the system of bodies m, n, o, reversed in 

 such a manner that FA", instead of being horizontal, shall become 

 vertical, &c. ; it is apparent, that in order to find the distance of 

 the resultant from the line FA' 1 , how vertical, it will be necessary 

 to take the sum of the moments with respect to FA", and to di- 

 vide this sum by the sum of the masses ; which gives 

 n ,, n m x A" A 4 n x B"B + o x C"C 



Cr Cr = - . 



m -\- n -\- o 



Having found the distance of the pointG from two fixed known lines 



Top. l. FA'', FO, the position of this centre G is evidently determined. 



It is here taken for granted that the distances A' A, A" A, B'B, 



BB', &c., are known, since the point through which FA", FC\ 



are drawn, is assumed at pleasure. 



77. If the distances A'' A, B"B, &c., are each zero ; that is, if 

 all the bodies are in the same straight line FA," the sum of the 

 moments with respect to this line is zero ; the distance G"G is 

 therefore zero. Accordingly, if several bodies, considered as points, 

 are in the same straight line, their common centre of gravity is also 

 in this line. 



78. If the lines FA' 1 , FC f , are either of them drawn in such 

 a manner as to have bodies situated on each side of it, instead 

 of the sum of the moments, we should say the sum of the mo- 

 ments that are found on one side, minus the sum of the moments 

 that are found on the other side. As to the denominator of the 

 fraction which expresses the distance of the centre of gravity, it 

 will always be composed of the sum of the masses, since all the 

 forces, by the nature of gravity, act in the same direction. What 

 is here said is applicable to any number of bodies, which, being 

 considered as points, are situated hi the same plane. 



