Centre of Gravity in particular Bodies. 65 



arid to the first ; but we shall confine ourselves for the present 

 to those solids, of which the parallel laminae have their respec- 

 tive centres of gravity all in the same straight line, as pyramids, 

 and solids of revolution. 



112. We begin with pyramids. Let h denote the height AC 

 of any pyramid^ x the perpendicular distance AP of any lami-pig. 4 

 na from the vertex ; c 2 the surface of the base ; we shall have 

 the surface of the lamina situated at the distance x from the ver- 

 tex, by this proportion Geom. 



409 - 



we have, therefore, 



c* 



A* 



whence /> <J (h x) d x becomes 



| 





But the pyramid which has x for its height, and for its base Geom. 



c* x 2 c a x 3 416> 



tf or is = - ' ; the distance of the centre of gravity 



therefore, is 



c x 



or A (4 h 3 a?) ; 



this quantity, when x = h = AC, is reduced to \ ft, and we have 

 the height CG" of the centre of gravity G, above the base = 1 h. 

 Now let G' represent the centre of gravity of the base ; the 

 line AG' will pass through G, the centre of gravity of the pyra- 

 mid ; and the parallels G"G, G X C, will give 



G"C or i h :. AC or h : : GG' : AG' ; 



whence GG 7 = 1 AG' which confirms what we have before said, 99 

 and shows that the centre of gravity of every pyramid, is one 

 fourth of the distance from th centre of gravity of the base, to 

 the vertex. 



Mech. 9 



