1G4 CENTRE OF GRAVITY. 



Guldinusa Properties. 



144. If any plane figure revolve about an axis //////</ t 'n /V.v 

 plane, the content of the solid generated ly this figure in re- 

 : ,i'<nnjh nni/ angle is equal to a prism, of //7//V/< t/tti 

 base is the revolving figure and height the length of (/< 

 described by the centre of yrnrt'ty of the area if th> //A////- 

 figure. 



The axis of revolution in this and the following proposition 

 is supposed not to cut the generating curve. 



Let the axis of revolution be the axis of x, ami the 

 plane of the revolving figure in its initial position the plane 

 of (x, y}; let fi be the angle through which the : 

 revolves. 



The elementary area A#Ay of the plane figure in revolving 

 througli an angle A0 generates the elementary solid whose 

 volume is y&O&x&y ; therefore the whole solid 



The limits of x and y depend on the nature of the curve. 

 But if y be the ordinate to the centre of gravity of the plane 

 figure, then, by Art. 118, 



the limits being the same as before. 



Therefore the whole solid = #(/// '/./ </// = yPffdx ay = the* 

 arc described by the centre of gravity multiplied by tin 

 of the figure, 



If any figure revolve about an axis lying in its own j-1<n 

 the surface of the solid generated is equal in area to the r& 

 angle, of which the sides are the length of the perimeter o 

 generating figure and the length of the path of the centr( 

 gravity of the perimeter. 



The surface generated by the arc As of the figure revolving 

 through an angle A0 is yA0 As ; therefore the whole surface 





