This derivation of the Galilean transformations is similar to that of the Lorentz transformations *here*.

Since space and time are assumed to be homogeneous, the transformations must be linear. The most general linear relationship is obtained with four constant coefficients: A, B, C, and D:

*x*′ = A*x* − B*t*

*t*′ = Ct − D*x*

Without loss of generality, if *t* = 0, let A = 1, and if *x* = 0, let *C* = 1. Then

*x*′ = *x* − B*t*

*t*′ = t − D*x*

The inverse transformation for the position from frame R′ to frame R should have the same form as the original but with its motion in the opposite direction, as is confirmed by algebra:

*x = x′* + B*t′*

*t = t′* + D*x′*

Then B = (*x* − *x*′) / *t*, which is the relative velocity V, and D = (*t* − *t*′) / *x*, which is the relative lenticity W.

*x = x′* + V*t′*

*t = t′* + W*x′*

If motion is along the *x* axis, then *t* = *t*′ and W = 0, and if it is along the *t* axis, then *x* = *x*′ and V = 0. In the former case

*x’ = x* − V*t*

*t’ = t*

and in the latter case

*x’ = x*

*t’ = t* − W*x*

These are the Galilean transformations.

*Revised 2021-06-11.*