130 IRE OF ( 



The centre of gravity of the solid is obviously in the straight 

 line Ox, so that we only r< <|uir- the value of x in order t 

 line its position. 



LSB, I. . 1. Let it be required to find the centre of 



ty of a portion of a paraboloid. Suppose v* 

 the equation to the generating parabola, and that the solid 

 is bounded by planes distant c and h respectively from the 



x; then 



2 A'-c 



If we put c = we find for the centre of gravity of a seg- 

 ment of a paraboloid commencing at the vertex 



- 2h 



Ex. 2. Required the centre of gravity of a portion of a 

 sphere intercepted between two parallel planes. 



Let y 1 = a* a? be the equation to the generating circle ; 



If we put c = and h = a, we find for the centre of gravity 

 of a hemisphere 



o; = {a. 



Ex. 3. Find the centre of gravity of the solid gener 

 by the revolution of the cycloid y = *J(2ax x f ) -f a vers" 1 - 



round the axis of x. 

 Here 



. o */ 



Now y* = 2ox - x* + 2a V(2aa - a*) vers" 1 - + a* fvers" 1 - ) . 



a \ a/ 



Thus the numerator of x consists of three integrals of 

 which we will give the values ; these values may be obtained 



rp 



without difficulty by transforming the integrals where vers' 1 : 



w 



