I'm looking at the 1927 paper of Thomas, The Kinematics of an Electron with an Axis, where he shows that the instantaneous co-moving frame of an accelerating electron rotates and moves with some infinitesimal velocity. He states:
At $t=t_0$ let the electron have position $\mathbf{r}_0$ and velocity $\mathbf{v}_0$, with $\beta_0=(1-{\mathbf{v}_0}^2/c^2)^{-\frac 1 2}$, in $(\mathbf{r}, t)$. Then, by (2.1), that definite system of coordinates $(\mathbf{R}_0, T_0)$ in which the electron is instantaneously at rest at the origin and which is obtained from $(\mathbf{r},t)$ by a translation and a Lorentz transformation without rotation is gven by $$\begin{align*}\mathbf{R}_0 &= \mathbf{r} - \mathbf{r}_0 + (\beta_0 - 1)\frac{ (\mathbf{r}-\mathbf{r}_0)\cdot \mathbf{v}_0}{{\mathbf{v}_0}^2}\mathbf{v}_0-\beta_0 \mathbf{v}_0(t-t_0)\tag{3.1a}\\ T_0 &= \beta_0\left( t - t_0 - \frac {(\mathbf{r} - \mathbf{r}_0)\cdot \mathbf{v}_0}{c^2}\right)\tag{3.1b}\end{align*}$$
By eliminating $(\mathbf{r},t)$ from equations (3.1) and the similar equations for $(\mathbf{R}_1, T_1)$,$$\begin{align*}\mathbf{R}_1 &= \mathbf{R}_0 + \frac{(\beta_0 - 1)} {{\mathbf{v}_0}^2}(\mathbf{R}_0\times (\mathbf{v}_0\times \mathbf{dv}_0)) - \beta_0 T_0(\mathbf{dv}_0 + (\beta_0 - 1)\frac{(\mathbf{v}_0\cdot \mathbf{dv}_0)}{{\mathbf{v}_0}^2}\mathbf{v}_0)\tag{a}\\ T_1 &= T_0 - \frac {\beta_0} c^2((\mathbf{R}_0\cdot(\mathbf{dv}_0 + (\beta_0 - 1)\frac{(\mathbf{v}_0\cdot \mathbf{dv}_0)}{{\mathbf{v}_0}^2}\mathbf{v}_0))) - d\tau_0\tag{3.3b} \end{align*}$$
What are the steps to get from (3.1) to (3.3)?
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