1G2 IKE OF c;i:.\\ i TV. 



Thus the centres of gravity of nil pl.-f f n 



cylinder are situated on a straight line parallel to the 

 rating lines of the evlin<; 



It' a portion of a cylinder be cut off by two pi. 

 neither of which is perpendicular to the axis, we may sup- 

 pose it to be the difference <t two portions which ha\ 

 their common base a section perpendicular t<> the axis. The 

 difference of the straight lines drawn from the eentr 

 gravity of the oblique sections perpendicular to the ortho- 

 gonal section will be the straight line joining those centres 

 of gravity. Hence the volume of a portion of a cylinder 

 contained between any two planes is equal to the pro- 1 net 

 of the area of an orthogonal section by the straight line 

 joining the centres of gravity of the oblique sections. 



143. Tlirovgh the centre of gravity of each face of a 

 tetrahedron a force acts at right angles to the face, and />/<>- 

 portional to the area of the face: if the forces all act inwards 

 or all act outwards they will be in equilibrium. 



Let A, JB, C, D denote the angular points of the tetrahe- 

 dron. The force acting on the face ABC, at its centre of 

 gravity, may be replaced by three equal forces acting at ; 



s to the face at the points A, B, C respectively. Simi- 

 lar substitutions may be made for the other forces. Thus we 



. acting at the point A, three forces resj;-ctiv.-ly at 

 angles to the three faces which meet at A and proportional to 

 the areas of those faces ; and. by what has been shewn in the 

 Propositions at the end of Chapter v. these three force 

 equivalent to a single force acting at A in the direction 

 icular to the face BCD, and proportional to the. ai 

 that lae.-. llmef. l,y Proposition I. at the end of Chaj 

 the proposed system of forces will be in equilibrium. 



The preceding result may now be extended to the following 

 proposition: Through the centre of gravity of each foci 

 polyhedron a force acts at //'/ .''< an</ 



portion"/ t the area of th< \f the forces all act in 



// act outtcni-'l* tl'ij u'itt be in i im. 



For each face of the polyhedron may be divided into 

 triangles; and the force, acting at the centre of gravity of 



