UNIVE. 



47.] DETERMINATION OF CENTROIDS. 25, 





46. For a homogeneous pyramid or cone, we have found in 

 Art. 28 (c) that the centroid lies on the line joining the vertex 

 to the centroid of the area of the base, at a distance equal to 

 of this line from the base. This is, of course, easily shown 

 directly by resolving the pyramid or cone into plane elements 

 parallel to the base, in a manner analogous to that used for the 

 triangular area in Art. 32. 



47. It may, perhaps, be well to formally state the principal 

 laws of symmetry for homogeneous solids, although they present 

 themselves so naturally that they are used almost instinctively. 

 For however simple and obvious these propositions may appear, 

 the beginner may be led into error if he does not use them 

 cautiously. The proof rests on the fundamental definition of 

 the centroid as a point such that for any plane through it the 

 sum of the moments is zero. 



(a) If the surface of the soiid have a plane of symmetry, i.e. a 

 plane such that every line perpendicular to it intersects the sur- 

 face in two points equidistant from the plane, the centroid lies 

 in this plane. Hence, the centroid of a homogeneous solid is 

 at once known if its surface possesses three planes of symme- 

 try. If the surface has two planes of symmetry, the centroid 

 lies on their line of intersection. 



(b) If the surface have an axis of symmetry, i.e. a line such 

 that every line perpendicular to '\\. intersects the surface in two 

 points equidistant from the line, the centroid must lie on this 

 axis. Two axes of symmetry in the same homogeneous solid 

 determine its centroid by their intersection. 



(c) If the surface have a centre, i.e. a point such that every 

 line through it intersects the surface in two points equidistant 

 from it, the centroid coincides with this centre. 



(d) If the surface have a diametral plane, i.e. a plane bisect- 

 ing all chords that are parallel to a certain direction, the centroid 

 lies in this plane. 



