CENTRE OF <ii:AVH V. 



Ex. 3. TV EMM nml tin .v///W cr>,,1i nt nf tic M<//,/ 



farmed Inj the revolution of a < -mul // t<nj<i,t t its 



vertex. 



Art. 133 we have found - for tin- distance of ti 

 o 



of gravity of tlie arc of a eycl"id fpnn its vertex : and tlio, 

 whole length of the arc is Hu. Therefore the surface of the 

 solid generated is 



2?r x ~ x 8a ; that is "" 7ra f . 

 o o 



And in Art. 113 we have found that the distance of the c 



of gravity of the area included between the cycloid and its 



7 



base from the vertex is a ; and the area BO included is 



o 



37ra*. Hence the area of the portion which in the present 

 case revolves round the tangent is 4?* 3?ra 2 , that is TTO". 

 And the centre of gravity of this area may be shewn to be at a 



distance from the vertex. (See Ex. (2) of Art. 109.) There- 

 fore the solid content of the figure generated is 2?r - 7ra s , that 

 is TrV. 



EXAMPLES. 



1. Find the centre of gravity of five equal heavy par- 

 ticles placed at five of the angular points of a regular 

 hexagon. 



2. Five pieces of a uniform chain are hung at 



points along a rigid rod without wright, and their lower cuds 

 an in a straight line passing through one end of the rod; 

 find the centre of gravity of the system. 



3. A plane quadrilateral ABCD is bisected by the dia- 

 gonal AC, and the other diagonal divides AC into two parts 

 in the ratio ofp to q ; shew that the centre of gravity of the 

 quadrilateral hes in AC and divides it into two parts in the 

 ratio of 1p + q to p + 2/7. 



