A PARTICULAR METHOD IN CENTROIDS 221 



arc with its chord be rotated round O through a very small Z a into 

 the position A'DB', the Center of Mass being now G'; then Z 

 AMA'=AOA'=GOG'=a. This rotation has been given for con- 

 venience of the construction : the result is the same as if the very 

 small arc AA' were transferred into the position BB', 



In the limit when AA' is infinitely small GG' is parallel to 

 AB, the Center of Mass of AA' travels from A to B along AB 



GG'AA' 

 AB arc 



OGXa rXa 



%. e. 



0Q-- 



AB arc 



radius X chord 



arc 

 If ZA0G=^ this becomes 



6 

 For a semicircle this becomes 



IT 



0G= ':^. (1) 



(2) 



Cor. The Center of Mass of a sector, radius r and Z 2^, being 

 that of an arc radius §r, and Z 26 is given by: 



-r^. 2r sin 6 .q\ 



Distance from center= ( o) 



For a semicircular lamina this becomes — , (4) 



TT 



6 being in this case — • 

 ^ 2 



These last two might be investigated separately by the method 

 of this article, by transferring an infinitely small triangle standing on 

 an infinitely small arc of the circle and having its vertex at the 

 center. 



