Centre of Gravity in particular Bodies. 51 



88. Accordingly, every thing which we have said hitherto 

 upon the common centre of gravity of several bodies, considered 

 as points, is equally applicable to bodies of whatever figure, if 

 we take, in estimating the moments, instead of the distance of 

 each body, the distance of its particular centre of gravity. 



89. Hence, finally, if several bodies, of whatever figure, have their 

 particular centres of gravity in the same straight line, or in the same 

 plane; their common centre of gravity will, in the former case, be in 

 the given straight line, and in the latter in the given plane. 



Application of the Principles of the Centre of Gravity to particular 



Problems. 



90. Let AB be a straight line uniformly heavy. It will be pig. 30. 

 seen at once without the aid of any demonstration, that the mid- 

 dle point P, of its length will be its centre of gravity. But in 

 order to illustrate and confirm the theory of moments, developed 

 in the preceding articles, let us seek the centre of gravity accor- 

 ding to the principles of this method. 



We imagine this straight line divided into an infinite number 

 of points, of which P p represents one ; and that each is multi- 

 plied by its distance from a fixed point, as the extremity A for 

 example. We then take the sum of these products, and divide 

 it by the sum of the parts of which Pp is one, that is, by the 

 line AB. Accordingly, if we call AB, a ; AP, x ; we shall have 

 Pp = d x ; and the moment of Pp will be equal to x d x, which Cal. 7. 

 must be integrated to obtain the sum of the moments. This sum 



x 2 



therefore will be equal to ; and in order to have it for the Cal. 82. 



whole extent of the line, we must suppose x = a, which gives 

 for the entire sum of the moments. Dividing this by the sum 



ft ft 



a of the masses, we have or for the distance of the cen- 8 g 



tre of gravity from the point A. Thus, the centre of gravity of a 

 straight line, uniformly heavy, is its middle point, as was before 

 manifest. 



