FIKST AND SECOND MOMENTS 



29 



Consider the circle as made up of a large number of elementary 

 triangles OAB with common vertex at (Fig. 25). Since the alti- 

 tude of each of these triangles is the radius R of the circle, from 

 the preceding article the / for each with respect to the point is 



AB x R* , 



r or the entire circle, therefore, 



or, since ^AB = circumference = 2 irR, this becomes 



(24) T O = ^' 



If D denotes the diameter of the circle, then R= and we also have 



Y 



If XX and YY are two rectan- 

 gular diameters of the circle, and r 

 is the distance of any element of 

 area &A from their point of inter- 

 section (Fig. 25), then 



Hence ^V 2 A^4 



f*^ T T T FIG. 25 



-*o = -LY + *X' 



Since a circle is symmetrical about all diameters, we have I x Jj 

 Therefore the / of a circle with respect to any diameter is 



-r ~r -r *- fl 



or 



(27) 



TT& 



4 



7TJ) 4 

 "64" 



24. I for composite figures. When a plane figure can be divided 

 into several simple figures, such as triangles, rectangles, and circles, 

 the / of the entire figure with respect to any axis may be found by 

 adding together the /'s for the several parts with respect to this 

 axis. Thus, in Fig. 26 each area may be regarded as the difference 



