CHAPTER VI. 



GENERAL THEOREMS. 



98. Centre of Inertia. The centre of inertia of a system 

 of particles is the centroid of points, which are the positions of 

 the particles, for multiples, which are the masses of the particles ; 

 it is the point whose coordinates x, y, z relative to any frame are 

 given by the equations 



where m is the mass of a particle of the system at the point 

 (x, y, z), and the summation extends to all the particles. 



If p is the density of a body at a point (x, y, z), the coordinates 

 of the centre of inertia of the body are x t y, z, where 

 - _ fffxpdxdydz 

 ~ ffjpdxdydz 



with similar expressions for y, z obtained by interchanging x with y 

 and z respectively, the integrations extending throughout the body. 



Since the centre of inertia of a body small enough to be handled coincides 

 with its centre of gravity as defined in Statics, we shall denote it by the 

 letter G. 



99. Relative coordinates. Let the coordinates of the 

 position of a particle of mass m be x, y, z referred to any 

 system of axes, and let x, y, z be the coordinates of the centre 

 of inertia of the system of particles referred to the same axes, 

 then we may put 



x - x + # , y = y + y , z = z + z , 



and the quantities x , y*, z 1 are the coordinates of the position of the 

 particle relative to parallel axes through the centre of inertia. 

 From the definition of the centre of inertia it follows that 



the summations referring to all the particles. 



