136 CENTER OF GRAVITY 



In 86 we obtained for the center of gravity of a number of 

 particles the formulae 



x ~\ ' y 



(30) 



In the present instance these become 



I I px dxdy I I py dxdy 



x = - y = - -> 

 j (p dxdy \\p dxdy 



the sign of summation being replaced by an integration which is 

 to extend over the whole area of the lamina. 



If the lamina is uniform, the value of p is constant, so that 



/ / px dxdy = p I lx dxdy, 

 and so on, and on dividing throughout by p, the formulae reduce to 



I lx dxdy I \y dxdy 



-X JA. _ , y^lLl __ 



I I dxdy I I dxdy 



106. Center of gravity of a solid. To find the center of gravity 

 of a solid we divide it into small solid elements by three systems 

 of planes parallel to the three coordinate planes. The volume of 

 any small element is then dxdydz, and its mass is p dxdydz. The 

 formulas of 86 now give the coordinates of the center of gravity 

 in the form 



I I I px dxdydz 1 1 1 ?& dxdydz 



_ = JJJ - _ ? y = JJJ -, etc. (31) 



I I ip dxdydz I / I p dxdydz 



