Page:Elementary Text-book of Physics (Anthony, 1897).djvu/47

system. The point thus determined is independent of the order in which the particles are taken into combination; it is a unique point, and depends only on the positions of the particles and their masses.

The centre of mass may be defined analytically as follows: Let the particles $m_{1},m_{2},...$ be referred to a system of rectangular coordinates. The coordinates $\xi ,\eta ,\zeta$ of the centre of mass are then given by the equations

$\xi ={\frac {m_{1}x_{1}+m_{2}x_{2}+...}{m_{1}+m_{2}+...}}={\frac {\sum mx}{\sum m}};$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$ $(21)$
$\eta ={\frac {m_{1}y_{1}+m_{2}y_{2}+...}{m_{1}+m_{2}+...}}={\frac {\sum my}{\sum m}};$
$\zeta ={\frac {m_{1}z_{1}+m_{2}z_{2}+...}{m_{1}+m_{2}+...}}={\frac {\sum mz}{\sum m}}.$

These equations are evidently consistent with the former definition of the centre of mass, if we remember that if the line joining any two particles be projected on one of the axes, the segments into which it is divided by the centre of mass of the two particles will be in the same ratio after projection as before. Consider the two particles $m_{1}andm_{2}$, and denote the coordinate of their centre of mass by $\xi$. Then from the former definition of the centre of mass we have $m_{1}(\xi -x_{1})=m_{2}(x_{2}-\xi )$, from which $\xi ={\frac {m_{1}x_{1}+m_{2}x_{2}}{m_{1}+m_{2}}}$. This demonstration can easily be extended to include all the particles of the system.

If some of the particles of the system be in motion, the centre of mass will, in general, also move. Its velocity is determined by the velocities of the separate particles. Let $\xi _{0},\eta _{0},\zeta _{0}$ represent the coordinates of the centre of mass at the time $t_{0}$, while $\xi ,\eta ,\zeta$ represent its coordinates at a later time $t$. The component velocities of the centre of mass are then given by the limit of the ratios ${\frac {\xi -\xi _{0}}{t-t_{0}}},{\frac {\eta -\eta _{0}}{t-t_{0}}},{\frac {\zeta -\zeta _{0}}{t-t_{0}}}$. Using the equations which define the coordinates of the centre of mass, we have: