16.] -MOMENTS AND CENTRES OF MASS. 7 



According as the mass is of one, two, or three dimensions, a 

 single, double, or triple integration over the whole mass will in 

 general be required for the determination of the moments 

 (xdM, \ydM % (zdM of the mass with respect to the co-ordi- 

 nate planes, as well as of the total mass (dM=M. 



Thus, for a mass distributed along a line or a curve we have, 

 if ds be the line-element, 



dM=pds; 



for a mass distributed over a surface-area we have, with dS as a 

 surface-element, 



finally, for a mass distributed throughout a volume whose 

 element is dV> 



If the mass be distributed along a straight line, the centroid 

 lies of course on this line, and one co-ordinate is sufficient to 

 determine the position of the centroid. In the case of a plane 

 area, the centroid lies in the plane and two co-ordinates deter- 

 mine its position ; we then speak of moments with respect to 

 lines, instead of planes. 



16. If the mass be homogeneous (Art. 8), i.e. if the density p 

 be constant, it will be noticed that p cancels from the numerator 

 and denominator in the equations (2), and does not enter into 

 the problem. Instead of speaking of a centre of mass, we may 

 then speak of a centre of arc, of area, of volume. The term 

 centroid is, however, to be preferred to centre, the latter term 

 having a recognised geometrical meaning different from that of 

 the former. 



The geometrical centre of a curve or surface is a point such 

 that any chord through it is bisected by the point ; there are 

 but few curves and surfaces possessing a centre. 



