HO 



THE OP <;I:A\ ITV. 



Curve. 



132. Suppose a circle of variable radius to move so that 

 its centre describes a given curve and IN plane is always 

 perpendicular to the tangent line of the curve, we may require 

 the centre of gravity of the solid generated. The .-in 

 case is that in which the radius is constant and the solid <>t' 

 in density; the result depends solely on. the nature of 

 the curve described by the centre of the circle, and for short- 

 ness the process is called finding the centre of grart'ty <f a 



Let BPQE be a plane curve ; BP the length measured 

 from some fixed point B, 

 x, y the 



BP=s, 



co-ordinates of P. Let k de- 

 note the area of a transverse 

 section ; then the volume of 

 the element PQ is &As, and 

 the co-ordinates of its centre 

 of gravity are ultimately x 

 and y. Hence 



_ fkxds fxds . f . . 



x = f j mj _ = Vj- ... (1) if A; be constant, 

 jas 



~dx 



.... (3). 



17FF 



From the equation to the curve y and ^ are known in 



