Centre of Gravity. 4.9 



Now if we imagine the figure reversed, the plane XZ becom- 

 ing vertical, and ZV horizontal, it will be seen that in order to 

 determine the distance E'G', corresponding and equal to EG, the 

 distance sought, it is necessary, according to the method above 

 pursued, to take the sum of the moments with respect to Z.F, as 

 if the bodies were all in the plane ZF, and to divide this sum by 

 the sum of the masses ; we have then every thing that is requi- 

 site in order to fix the position of the centre of gravity. 



83. Hence, by recapitulating what we have said, this prob* 

 lem reduces itself to the following particulars ; 



(1.) When the several bodies, considered as points, are situated 

 in the same straight line, we take the sum of the moments with Fig. 29* 

 respect to a fixed point F, assumed arbitrarily in this line, and 

 divide this sum by the sum of the masses, and the quotient will 

 be the distance of the centre of gravity G from the point F. 



(2.) When the several bodies, considered as points, are all in 

 the same plane ; through a point -F, taken arbitrarily in this Fj g> 07. 

 plane, we suppose two lines FA", FC', to be drawn at right an- 

 gles to each other ; and having let fall perpendiculars upon each 

 of these two lines from each gravitating point, we imagine that 

 these gravitating points are applied successively to the lines FA", 

 FC, where their perpendiculars respectively fall. We then 

 seek, as in the case just stated, what would be the centre of 

 gravity G" in FA", and what would be the centre of gravity 

 G' in FO ; drawing lastly through these two points the lines 

 G"G, G'G, parallel respectively to FO, FA", and their point of 

 meeting G will be the centre of gravity sought. 



(3.) When the several bodies, considered as points, are in 

 different planes, we imagine three planes, one horizontal, and Fi 8- 23 - 

 the two others vertical and perpendicular to each other. From 

 each gravitating point we suppose perpendiculars let fall upon 

 each of these three planes ; we then take the sum of the moments 

 with respect to each plane, and dividing each of these sums by 

 the sum of the masses, we shall have the three distances of the 

 centre of gravity from the three planes respectively. 



84. It must be recollected, moreover, in what is above said, 

 that when the bodies are on different sides of the line or plane 

 with respect to which the moments are considered, it is necessa- 



Meck. 7 



