I have seen in some texts that the Galilean transformation of the magnetic field across inertial reference frames is given by: $\vec B'= \vec B - (1/c^2)(\vec v\times \vec E)$. Every where it is stated that it can be derived using Galilean transformation. However I have never found it derived using only Galilean transform. (i.e. without taking help of Lorentz transformations). The other equation concerning Galilean transformation of electric field: $\vec E'=\vec E + (\vec v \times \vec B)$ has been seen derived by only Galilean transformation (i.e. by invariance of Lorentz force) very elegantly in many texts. Is it possible to derive this equation $\vec B'= \vec B - (1/c^2)(\vec v\times \vec E)$ by only using Galilean transformation?