Centre of Gravity in particular Bodies. 57 



will be easy, by the method of moments, to find the centre of 

 gravity of any body whatever. 



101. Such is the general manner of finding the centre of grav- 

 ity of bodies, the parts of which are independent of each other, 

 or rather when we have not the expression of the law by which 

 they are connected together. 



But when the parts of a figure or body have a relation that 

 can be expressed by an equation, the centre of gravity may be 

 found much more readily. 



102. Let it be required to find the centre of gravity G, of any Fig. 40 

 arc of a curve AM ; we imagine an infinitely small arc M m, and 

 take for the. axis of the moments any line GAT, parallel to the 

 ordinates, which are supposed to be perpendicular among them- 

 selves. Suppose, moreover, that the distance of C from the ori- 



gin A of the abscissas = h, h being taken of any magnitude at 

 pleasure. To obtain the distance GG' of the centre of gravity 

 from the axis G/V, we must take the sum of the moments of the 75. 

 arcs Mm, and divide it by the sum of the arcs M m ; that is, by 

 the arc AM. Now the arc Mm being infinitely small, the dis- 

 tance of its middle point n, from the straight line CJV, may be 

 considered as equal to MN. We shall have, therefore, 



Mm X MN 



for the moment of this infinitely small arc. But, calling AP, #, c a ] 97. 

 and PM, y, we shall have Mm ^/dx 2 + di/ a , and 



MN = PC = h x ; 



therefore (h x) \/dx* + dy* is the moment of the arc Mm, 

 and consequently f (h x) v'dx 2 -f- ay*, or the integral of 

 (h x) v dx* + d y 2 , is the sum of the moments of all the infi- 

 nitely small arcs Mm, of which the arc AM is composed. We 

 have, therefore, 



AM 



With respect to the arc AM, which is a divisor in this quantity, 

 we have given a method of determining it exactly, when that canCal. 96 

 be done; and another method of determining it by approxi- Ca j llo 

 mation. 



Mech. 8 



