Centre of Gravity in particular Bodies. 55 



centre of gravity ; therefore this centre is at the intersection G 

 of AH and IE. 



97. By pursuing the same kind of reasoning which we adopted 

 in the case of the triangle, it might be demonstrated that the 

 point G is the centre of gravity of the surface of a regular 

 pentagon. 



In general, it may be shown, by the same method, that the 

 centre of gravity of the perimeter, as well as of the surface of any 

 regular polygon, of an odd number of sides, is the point of in- 

 tersection of two straight lines, each of which is drawn from the 

 vertex of one of the angles' to the middle of the opposite side. 

 And when the number of sides is even, the centre of gravity, Fig. 37. 

 both of the perimeter and of the surface, is the point of intersec- 

 tion of two straight lines drawn through the middle points of two 

 pairs of opposite sides. We might also extend this mode of rea- 

 soning to the circle, by regarding it as a polygon of an infinite 

 number of sides, and we should find that the centre of gravity 

 of the circumference, and of the surface, is the centre. 



When the number of lines, surfaces, bodies, &c., is not con- 

 siderable, the centre of gravity may be found by the method of 

 articles 53, 54. Let the three points .#, B, C, for example, be the Fig. 38 

 centres of gravity of three lines, or three surfaces, or three 

 bodies, whose weights are represented by the masses m, n, o. 

 Having joined two of these points^ as B and C, by the line BC^ 

 we divide BC at G', in such a manner as to give 



n : o : : CG' : BG' 

 or 



n + o : n : : CB : CG' ; 



and the point G' thus found, will be the centre of gravity of the 

 two weights n, o. We now draw G'A, and supposing the two 

 masses w, o, united in G', we divide, in the same way, G'A in the 

 inverse ratio of the two masses m and n + o, that is, so as to 

 give 



n + o : m : : JIG : GG, 

 or 



n + o + m : m :: AG : GG ; 



and the point G will be the common centre of gravity of the 

 three weights m, w, o. We might proceed in a similar manner 

 with a greater number of bodies. 



