The sum of the products of the mass of each of the points by the kth power of its distance to ($\Delta$), $(d)$ or $(D)$ is called the kth order moment of a given system with respect to a plane ($\Delta$), to a line $(d)$ or to a point $(D)$.
$\sum_{i=1}^{n} r_{i}^{k}m_{i}$
From this point of view, the moment of order zero is, quite simply, the total mass
$m=\sum_{i=1}^{n} m_{i}$
The first order moment with respect to a plane ($\Delta$) is called the static moment of the system with respect to this plane. The static moments with respect to the coordinate planes enter the expressions giving the coordinates of the center of gravity of a system
$x_{g}=\frac{\sum_{i=1}^{n} m_{i}x_{i}}{m}\;\;,y_{g}=\frac{\sum_{i=1}^{n} m_{i}y_{i}}{m}\;\;,z_{g}=\frac{\sum_{i=1}^{n} m_{i}z_{i}}{m}$
In this case, the distances $x_{i},y_{i},z_{i}$ to the plans containing the coordinate axes are algebraic quantities, i.e. they can be positive or negative.
The second order moments are usually called the moment of inertia of the system. Thus, the expressions
$\sum_{i=1}^{n} x_{i}^{2}m_{i}$, $\;\;\;\sum_{i=1}^{n} y_{i}^{2} m_{i}$, $\;\;\;\sum_{i=1}^{n} z_{i}^{2}m_{i}$
are the moments of inertia of the system with respect to the planes containing the coordinate axes, the expressions :
$\sum_{i=1}^{n} (y_{i}^{2}+z_{i}^{2})m_{i}\;\;,\sum_{i=1}^{n}(z_{i}^{2}+x_{i}^{2})m_{i}\;\;, \sum_{i=1}^{n} (x_{i}^{2}+y_{i}^{2})m_{i}$
are the moments of inertia with respect to the axes $(OX),(OY),(OZ)$, finally the expression
$\sum_{i=1}^{n} (x_{i}^{2}+y_{i}^{2}+z_{i}^{2})m_{i}$
is the moment of inertia with respect to the point O .
Moreover, it happens that we have to deal with expressions
$\sum_{i=1}^{n} y_{i}z_{i}m_{i}\;\;\;\;,\sum_{i=1}^{n} z_{i}x_{i}m_{i}\;\;,\sum_{i=1}^{n} x_{i}y_{i}m_{i}\;\;,$
which are called centrifugal moments of the system with respect to the axes $(OX),(OY),(OZ)$.
If we find ourselves in the presence not of systems with a finite number of points, but of masses distributed in a continuous way, the previous sums are replaced by integrals , simple, double and triple, according to whether the masses are distributed on a line, surfaces or volumes, it is necessary instead of the factor $m_{i}$ , to introduce the product of the density $f(M)$ at the point M by the element of length, of surface or of volume. 374