THE CENTROID 209 



The two areas a v and 2 can now Dc replaced by a single area 

 (a l + flj) situated at the point M, and this can now be combined 

 with the third area a 3 , the centroid of this system can be taken 

 to be a point N on the line MC, and if the co-ordinates of N are 

 /7a- 



Then (a 1 + a 2 )z l 



= a l x 1 



a x + a 2 4- a 3 



This fraction is such that the numerator is the sum of the 

 moments of the areas about the axis OY and the denominator is 

 the sum of the areas. The effect of introducing to the system 

 another area a 4 , situated at a point whose co-ordinates are x 4 , y 4 , 

 is to increase the numerator by the moment of that area, and at 

 the same time to increase the denominator by that area. 



This process must be continued until all the small areas which 

 make up the total area are taken into account 



or Ax = 2ax. 



Working with the ordinates y v y 2> y s , etc., it can be shown in 

 a similar manner that 



- _ 



or A.y = ^ay 



The relations A,r = 2o#, and Ay = 2ay enable us to deter- 

 mine the position of the centroid P of the whole area. 



116. Let P be the position of the centroid of the irregular area 

 (Fig. 52) the co-ordinates of P being x, y. 



Let this area rotate about the axis of x, describing a surface of 

 revolution. 



Taking the whole area to be built up from a system of small 

 areas a lt a z , a 3 , a 4 , etc., situated at points A, B, C, D, the 

 co-ordinates of these points being (x-^y^), (x^j^, (x<gj^, (# 4 t/ 4 ), 

 then each of these small areas will describe an elementary 

 ring, the area a x describing a ring of volume 27ta 1 t/ 1 , the area a z 

 one of volume 2Ttag/ z , and so on. The volume of the whole 

 surface of revolution will be the sum of the volumes of all of 

 these elementary rings. 



Then V ox = 27ra 1 t/ 1 + 2710,^2 + 27WZ 3 t/ 3 + . . . 



