Centre of Gravity in particular Bodies. 59 



and consequently, 



Wax x* CA x MM' 

 = a 



MAM * MAM.' MAM 



which gives this proportion, 



MAM' : MM' : : CA : CG. 



Thus we obtain the following rule ; that the, distance of the 

 centre of a circle from the centre of gravity of any arc of this circle, 

 is a fourth proportional to the length of the arc, its chord and radius. 



These formulas may be applied to any other curve. 



We pass now to the consideration of the centre of gravity 

 of plane surfaces bounded by curved lines. 



104. Let it be required to find the centre of gravity of the Fig. 42. 

 surface APM; and let G represent this centre. In order to ob- 

 tain the distance GG', it is necessary to take the sum of the 

 moments of the small trapezoids MP pm, with respect to CJV, 

 and to divide this sum by the sum of the trapezoids, that is, by the 

 surface APM. Now the centre of gravity F of this small trapezoid 

 must be in the middle point of the straight line n K, equally dis- 

 tant from MP and nip, which point we can suppose to be in MP, 

 on account of the infinitely small height Pp. We shall have, 

 therefore, FL = CP ; and the moment oiPpmM will be 



Pp m M X CP, 



that is, (hx)ydx, calling always CA, h, and AP, x. There* 

 fore the sum of the moments will be f (h x) y d x, and conse- 

 quently the distance 



rr 1 



APM 

 It will be found, likewise, that the distance 





105. The centre of any plane surface may be found, in the 

 same manner, by decomposing it into infinitely small trapezoids. 



For example, let the surface in question be the triangle ANN 1 , 

 and take the base NN' and the height AC for the axes of the 

 moments ; now calling AP, x, MM', y, and AC^ h, we shall have l * 43 ' 

 MM m' m y d x ; and the moment of this trapezoid, with res- 



