the three-dimensional vector space. In the three-dimensional vector space $(\emph{\textbf{j}}_1 ,~\emph{\textbf{j}}_2 ,~\emph{\textbf{j}}_3)$, the operator is $\blacktriangledown = \Sigma \emph{\textbf{j}}_k \partial_k$, with $\emph{\textbf{j}}_k^{~2} = 1$.

In the gravitational field equations, the operator $\nabla$ should be substituted by the operator $\blacktriangledown$. Meanwhile $\textbf{b}$, $\textbf{g}$, and $\textbf{s}$ are substituted by $\textbf{b}^\prime = \Sigma b_k \emph{\textbf{j}}_k$, $\textbf{g}^\prime = \Sigma g_k \emph{\textbf{j}}_k$, and $\textbf{s}^\prime = \Sigma s_k \emph{\textbf{j}}_k$ respectively. The cross and dot products of quaternion are substituted by that of vector respectively. And the gravitational field equations, Eqs.(9)-(12), can be transformed into, $$\begin{aligned}
&& - \blacktriangledown \cdot \textbf{b}^\prime = 0 ~,
\\
&& \partial_0 \textbf{b}^\prime - \blacktriangledown \times \textbf{g}^\prime / v_0 = 0 ~,
\\
&& \blacktriangledown \cdot \textbf{g}^\prime / v_0 = - \mu_g s_0   ~,
\\
&& \blacktriangledown \times \textbf{b}^\prime + \partial_0 \textbf{g}^\prime / v_0 = \mu_g \textbf{s}^\prime ~,\end{aligned}$$ where $\textbf{b}^\prime = \blacktriangledown \times \textbf{a}^\prime$, $\textbf{g}^\prime / v_0 = \partial_0 \textbf{a}^\prime + \blacktriangledown a_0$. $\textbf{a}^\prime = \Sigma a_k \emph{\textbf{j}}_k$.

Expressing the above into the scalar equations will reveal that Eqs.(9)-(12) and Eqs.(20)-(23) are equivalent to each other. And the definition of Eq.(6) requires to substitute $\textbf{g}^\prime$ by $\textbf{g}^{\prime\prime} = - \textbf{g}^\prime$, in order to approximate to the Newton’s law of universal gravitation in 