oiul vortical plane by a quantity a 1 " 



- D " ' &lld tllis P int Wil1 1>C tll celltre of 8- 

 vity of the thriven bodies, as is evident from the last Art. 



5. If a body be placed upon a horizontal plane, and a line drawn from 

 its centre of gravity perpendicular to that plane, the body will be sus- 

 tained or not, according as the perpendicular falls within or without the 

 base. 



6. If a body be suspended by a point, it will not remain at rest till the 

 centre of gravity is in the line which is drawn through that point per- 

 pendicular to the horizon. 



Cor. Hence to find the centre of gravity of any plane mechanically, 

 s uspcud it by a given point in or near its perimeter, and when it is at rest, 

 lra\v across it a vertical line passing through that point. Suspend it in 

 like manner by another point, and draw a vertical line as before. The 

 intersection of these lines is the centre of gravity of the plane. 



7. If any momenta be communicated to the parts of a system, its cen- 

 tre of gravity will move in the same manner that a body equal to the 

 sum of the bodies in the system would move, were it placed in that cen- 

 tre, and the same momenta communicated to it in the same directions. 



8. In any machine kept in equilibrium by the action of two weights, if 

 an indefinitely small motion be given to it, the centre of gravity of the 

 weights will neither ascend nor descend. 



9. Formulae for finding the centre of gravity of a body considered as an 

 area, solid, surface, or curve. 



Let x, y, and z t represent the abscissa, ordinate, and curve, D = dis- 

 tance of vertex from the centre of gravity ; then 



ft. ijxdx 

 For an area, D---^ . 



For a solid, D = '* * 



fl. -yxdz 



For a surface, D = -~ -3 . 

 fl. y d z 



fl. xdz 

 For a curve line, D -- 



Ex. 1. In a triangle and conical surface, let a be the liue from the ver- 



2 a 

 lex bisecting the base, then D -^ . 



53 C* 



