Page:The Meaning of Relativity - Albert Einstein (1922).djvu/31

Rh Multiplying by the velocity, a tensor of rank 1, we obtain the tensor equation

By contraction and multiplication by the scalar $$dt$$ we obtain the equation of kinetic energy

If $$\xi_\nu$$ denotes the difference of the co-ordinates of the material particle and a point fixed in space, then the $$\xi_\nu$$ have the character of vectors. We evidently have $$\frac{d^2x_\nu}{dt^2} = \frac{d^2\xi_\nu}{dt^2}$$, so that the equations of motion of the particle may be written

Multiplying this equation by $$\xi_\mu$$ we obtain a tensor equation

Contracting the tensor on the left and taking the time average we obtain the virial theorem, which we shall not consider further. By interchanging the indices and subsequent subtraction, we obtain, after a simple transformation, the theorem of moments,

It is evident in this way that the moment of a vector